An Overview of Sequential Bayesian Filtering in Ocean Acoustics

IEEE JOURNAL OF OCEANIC ENGINEERING

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An Overview of Sequential Bayesian Filtering in Ocean Acoustics Caglar Yardim, Member, IEEE, Zoi-Heleni Michalopoulou, Senior Member, IEEE, and Peter Gerstoft

Index Terms—Acoustic signal processing, acoustic tracking, ensemble Kalman filter, extended Kalman filter (EKF), ocean acoustics, particle filter (PF), sequential importance resampling (SIR), sequential Monte Carlo methods, unscented Kalman filter (UKF).

I. INTRODUCTION

A

Depending on the data and the problem, Bayesian approaches can be used to address inversion or tracking problems (a brief comparison is given in Appendix I). Sequential Bayesian filtering, tying together information on parameter evolution, a function relating acoustic field measurements to the unknown quantities, and a statistical description of the random perturbations in the field measurements, offers a suitable framework for the solution of such problems. When functions relating 1) parameters between consecutive steps and 2) data and parameters are linear and noise is additive and Gaussian, the Kalman filter (KF) is the optimal estimator in terms of minimizing mean squared error (MSE). The KF propagates expectations and covariances of the unknown parameters from step to step, fully characterizing posterior pdfs. Some early examples used in source tracking in the ocean with KFs are given in [16] and [17]. Nonlinear functions require variations or generalization of the standard filter. Implementation in ocean acoustic problems of a straightforward extension, the extended Kalman filter (EKF), which linearizes mildly nonlinear functions, has been pioneered in a series of papers [18]–[27]. More recently, unscented Kalman filters (UKFs) have achieved better root-mean-square (RMS) error and convergence performance than EKFs by selecting deterministic points called sigma points to represent parameter pdfs [28], [29]. For large parameter vectors, the ensemble KF (EnKF) is efficient [30]. Highly nonlinear systems and complex noise processes require numerical methods for the computation of posterior pdfs. These approaches are termed particle filters (PFs) and have been encountered in ocean acoustic applications in [29] and [31]–[33]. In this tutorial paper, we briefly review the foundations of sequential filters, starting from the well-known KF and its variants and proceeding with PFs. These techniques formulate sequential estimation using physical relations between unknown parameters and measurements embedded in noise environments of a diverse nature. We examine and compare filters and present examples, illustrating practical challenges and solutions. A more detailed presentation of these methods, including theoretical derivations, can be found in many excellent signal processing papers [34]–[39] and texts on sequential Monte Carlo methods [40]–[43]. Once basic principles and methods of sequential filtering are discussed, the focus is shifted to an overview of sequential filtering implemented for parameter estimation of dynamical systems in ocean acoustics. We present applications in target tracking, wave estimation, geoacoustic inversion, frequency, and arrival time tracking. The paper is organized as follows. First, a general background about the state-space formulation for the estimation of evolving

IE E W E eb P r Ve oo rs f ion

Abstract—Sequential filtering provides a suitable framework for estimating and updating the unknown parameters of a system as data become available. The foundations of sequential Bayesian filtering with emphasis on practical issues are first reviewed covering both Kalman and particle filter approaches. Filtering is demonstrated to be a powerful estimation tool, employing prediction from previous estimates and updates stemming from physical and statistical models that relate acoustic measurements to the unknown parameters. Ocean acoustic applications are then reviewed focusing on source tracking, estimation of environmental parameters evolving in time or space, and frequency tracking. Spatial arrival time tracking is illustrated with 2006 Shallow Water Experiment data.

common feature of inverse problems in ocean acoustics is that underlying physical parameters are estimated from measured acoustic data. Examples include source localization [1]–[4], geoacoustic inversion [5]–[9], and marine mammal signal processing [10]. In a Bayesian framework, prior knowledge and acoustic models are combined with a likelihood function to provide posterior probability density functions (pdfs) of parameters of interest. This formulation was first proposed in source localization [11], [12]. Geoacoustic inversion was subsequently approached in a similar fashion, estimating, in addition to source location, ocean environment parameters and their uncertainties [13]–[15]. Often, such parameters evolve in time or space, with acoustic data arriving online at consecutive steps. Information on parameter value evolution and uncertainty at preceding steps can be invaluable for the determination of future estimates but is frequently ignored.

Manuscript received September 02, 2010; revised November 21, 2010; accepted December 03, 2010. This work was supported by the U.S. Office of Naval Research Code 32, under Grants N00014-09-1-0313, N00015-05-1-0264, N00014-05-1-0262, and N00014-10-1-0073. Associate Editor: R. Chapman. C. Yardim and P. Gerstoft are with the Marine Physical Laboratory, Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093-0238 USA (e-mail: [email protected]; [email protected]). Z.-H. Michalopoulou is with the Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 071028 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JOE.2010.2098810

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Fig. 1. Sequential Bayesian filtering. From state , state is first pre. As data become available, meadicted via the state (1), providing . surement (2) is employed to update state , providing

In a typical ocean acoustic example, function represents some form of a forward or acoustic propagation model and consists of complex-valued acoustic data measurevector ments across a hydrophone array. A sketch of sequential localization of a moving target as described by (1) and (2) is shown in Fig. 1 with circles representing is the estimate of the target location at step . Term using data obtained up to target location and velocity at . By exploiting knowledge of target poand including step sition and trajectory , a filter first predicts , the target location and velocity at step . The prediction is then upbecome available. dated when new data


parameters in dynamical systems is given in Section II together with the basics of Bayesian filtering. The KF framework is introduced in Section III together with its extensions such as the EKF, the UKF, and the EnKFs. The PFs are presented in Section IV. The filter equations are derived starting from the basic importance sampling (IS) concepts, moving to sequential IS, finally deriving the commonly used PF, often referred to as sequential importance resampling (SIR). Advanced PFs are briefly mentioned in Section IV and are presented more extensively in Appendix II. Practical challenges arising in sequential Bayesian filtering are discussed in Section V and ocean acoustic applications are discussed in Section VI. Although Section VI is dedicated to solving estimation problems in the ocean, specific issues in ocean acoustics as they relate to filtering are discussed throughout the paper. II. BACKGROUND

A. State-Space Model

In dynamical systems, a major goal is to estimate parameters that evolve sequentially with time or space. As data become available, the unknown parameters forming a state vector are estimated sequentially using collective data history and prior knowledge on evolution of the state. Tracking in time or space presents us with problems that can be readily formalized in the following framework. be the measurement vector (for example, pressure Let along a hydrophone array in ocean acoustics) at step and let represent the state vector (for example, location of a source emitting signal ), where . The size of the state vector is . Our goal is to sequentially estimate as data meabecome available. surements Two equations define a state-space model (1) (2)

The state equation (1) describes the evolution or transition of with and assumes that states follow a first-order Markov process. Function is a known function relating the state vector at step to that at step . Variable is the process or state . noise and has a known pdf The measurement equation (or observation equation) (2) reto state vector through a known funclates measurements tion . Variable is the measurement noise with a pdf . and can be additive, multiplicative, or incorpoNoise rated in the state and measurement through complex functions and of and , respectively. In addition to the state vector data , the functions and , and noise components and can also change with . In most ocean acoustic applications, the noise terms are adare independent of . Hence, (1) ditive and functions and and (2) are simplified to (3) (4)

B. Bayesian Filtering

Examining the problem from a Bayesian standpoint, we are rather than simple interested in deriving the posterior pdf for of the state vector is aspoint estimates. The initial pdf sumed to be known; it can be assumed in a common form such as uniform or Gaussian, based on ground-truth information, or estimated using numerical techniques such as a Markov chain Monte Carlo (MCMC) sampler on the initial data [14], [44]. be the set of data observed at the Let be the sequence first steps and let of unknown state vectors. It is desirable to estimate the state . Due to vector and its uncertainty at all steps via pdf the computational cost, the problem is simplified by recursively from , estimating the marginal pdf . rather than estimating joint pdf With available, we can predict through the transition pdf . The transition density . Due to the is determined by the state (1) and noise pdf does first-order Markov chain assumption of , not depend on data . Density can be written as

(5)

When a new measurement becomes available, we can upusing the likelihood of the state vector date the state

(6)

YARDIM et al.: AN OVERVIEW OF SEQUENTIAL BAYESIAN FILTERING IN OCEAN ACOUSTICS

and the prediction pdf of (5). The new estimate of the state vector at is calculated via Bayes theorem

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TABLE I KALMAN FILTER

(7) The denominator in (7) is the normalizing constant of , termed the evidence, and may be expressed as (8)


contains all information necessary for the Density computation of marginal pdfs of elements of , moments, and modes. Optimal sequential estimators [45], such as the minimum mean squared error (MMSE) estimator, which calculates of the state after data have been the expected value observed, and the maximum a posteriori (MAP) estimator that provides the likely value of the state also after data measurement, can be readily implemented. III. KALMAN FILTERS

The discussion of Section II-A raises the question of how to track optimally an evolving density . Kalman [46] showed that, if a Gaussian distribution can represent this evolving pdf, the optimum Bayesian estimator (one that attains the Bayesian Cramér–Rao lower bound (CRLB) [47] with the MMSE) exists and can be computed analytically. This optimum sequential estimator has been termed the KF. Having a Gaussian posterior at each step implies the following. 1) State and measurement noise must be additive. Noise variand , and the prior must be Gaussian. ables and are known linear functions of the state and mea2) surement vectors. This reduces the system given in (1) and (2) to (9) (10)

where and replace and as matrices representing known linear functions of the state vector . State and measurement noise covariances and are

(11)

Even though the parameters given here are real, it is straightforward to extend these to complex cases. The requirements given above are fairly restrictive, however the resulting filter is straightforward to implement. Because a Gaussian pdf is uniquely and completely defined by its first two moments, mean and covariance, the KF needs only recomputing mean and covariance at step , using , , and the new measurement . This is a two-step procedure, repeated at each . • Predict: This stage predicts the current value of the state given its previous value using (9). This is represented by and .

• Update: This stage updates or corrects the values predicted in the previous stage given the information obtained from measurement . This provides the posterior at represented by and . The resulting set of equations for the linear, Gaussian KF is represents the Kalman gain. given in Table I [46], where A. Extended Kalman Filter

Although elegant and easy to implement, the KF has strict linearity and Gaussian pdf requirements that make it unsuitable for a large class of systems that cannot be characterized by (9) and (10). An obvious way to extend the KF framework is linearizing in (1) and (2) around the current state value. functions and and Gaussian noise When coupled with a Gaussian prior terms, the posterior will remain Gaussian. Thus, only mean and covariance need to be tracked via slightly modified KF equations. The new filter, the EKF [45], [48], linearizes locally the state and measurement equations using the first terms in the Taylor series expansions. For the EKF to perform well, nonlinearities should be small and the underlying densities should be close to Gaussian. However, even under these circumstances, because of the implied approximations, the EKF cannot claim the optimality enjoyed by the KF for linear-Gaussian systems. For large nonlinearities, the mean and covariances will not be mapped correctly as shown in the example of Section III-B. Nevertheless, the EKF has been implemented successfully in numerous applications in areas such as radar and sonar target tracking among others. The EKF algorithm is summarized in Table II [45], [48]. B. Unscented Kalman Filter

The simplicity of the Kalman framework arises from its Gaussian nature. The filter only needs to carry the mean and covariance to the next step. A nonlinear system will disrupt the flow of evolving Gaussian pdfs, because a Gaussian random variable passing through a nonlinear transformation will lose its Gaussian form and solely mean and covariance will not be adequate in fully defining this density. This realization leads to the development of KFs that use sigma points coupled with

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TABLE III UNSCENTED KALMAN FILTER. ( IS THE th COLUMN OF THE MATRIX SQUARE ROOT [49], [50])


TABLE II EXTENDED KALMAN FILTER

unscented transforms (UTs) [49]. The UT enables the propagation of the mean and covariance through nonlinear functions. KFs that make use of the UT are called UKF [50]. While the EKF enforces linearity through analytic linearization, the UKF enforces a Gaussian distribution while keeping the functional nonlinearity. This means that the filter assumes that any nonlinear transformation of a Gaussian input pdf will always be another Gaussian pdf. Since, under this assumption, a Gaussian input results in a Gaussian output, the system is effectively linearized in a statistical rather than analytical sense. This still enables the filter to carry all necessary information by propagating only the mean and covariance as required by the KF. In the UT, a number of weighted sigma points are carefully chosen so that they capture the mean and covariance of the state variable (the sigma points depend on the matrix square root of covariance ; see Table III). As the random variable undergoes a nonlinear transformation, these points are propagated through this nonlinear function and are used to reconstruct the new mean and covariance using the UT weights [49]. Hence, unlike with the EKF, mean and covariance to at least second order of the nonlinearity (third if the initial pdf is Gaussian) can be computed accurately. sigma The UKF uses a recursive formulation where points and their corresponding weights are generated. These are used with the UT to perform the mean and covariance calculations required in the Kalman framework. The UT weights are computed using three independent parameters: prior knowledge parameter , scaling parameter , and parameter , which controls the spread of the sigma points around the mean. The second scaling parameter can be com. The algoputed from these parameters: rithm is presented in Table III. Details about the derivation can be found in [49]. Although UKFs are derivative-free, an improvement over EKFs, and efficient relative to more advanced techniques to be

discussed later, they have two disadvantages as demonstrated in the example below. 1) Example: EKF Versus UKF Versus PF: Assume that a goes Gaussian random variable with with through a nonlinear transformation

A cloud of 10 000 particles drawn from is plotted in Fig. 2 together with the mean and covariance ellipse. When these particles are propagated through the nonlinear equation, the distribution becomes non-Gaussian as shown. The analytical linearization in EKF uses the Jacobian

This will result in a Gaussian density with a mean and covariance of and represented by and


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with a large covariance matrix in a high-dimensional system is computationally inefficient for the classical KF. This drawback is addressed by using an ensemble of points, similar to the particle set in a PF, to replace the covariance matrix with the ensemble covariance matrix using MC sampling. For detailed theory and practical implementation, see [55]. IV. PARTICLE FILTERS


Fig. 2. Mean and covariance propagation of the EKF and UKF in a nonlinear, non-Gaussian system with . True mean , means from analytical , and statistical linearizations are given with covariance estimates in ellipses for the true covariance and the estimates from the Jacobian terms of (EKF) and the UT (UKF).

Sometimes, the dynamics of a problem cannot be adequately captured by the Kalman family of filters, because of the nonlinear structure of state and measurement equations and the nature of the noise. When KFs and their extensions fail, the estimavia filtering posterior pdfs may be possible tion of using numerical sequential MC techniques, commonly referred to as PFs. using a cloud of parPFs track the posterior pdf that evolve with step . All ticles other estimates can be calculated from , including and solutions, the variance in the estimates, and of the th state variable the marginal posterior pdf

a dashed ellipsoid in Fig. 2. Both mean and covariance have high errors in their estimates. Mean and covariance are also obtained by using the sigma point generation and a successive UT given in Table III. The values obtained by UT are very close to their true values with almost identical ellipses. Fig. 2 highlights two main drawbacks of KFs in nonlinear, non-Gaussian problems. The first is the effect of nonlinearity on the accurate estimation of mean and covariance. This example is “too nonlinear” for the EKF but not for the UKF for obtaining good mean and covariance estimates. However, there are many cases where the nonlinearity may be severe and may require an even higher order accuracy than the UKF can provide to correctly capture mean and covariance. The other is that the densities may be too non-Gaussian to be represented using the first two moments, even if those can be cannot be sufcalculated correctly. The non-Gaussian pdf ficiently characterized with mean and covariance alone in this example. In such cases, the only way to accurately propagate complex, non-Gaussian pdfs is to store the cloud of particles that represent the pdf as shown in Fig. 2. This forms the basic philosophy of particle filtering. The UKF is the first and most widely used sigma-point KF. It is possible to select different sigma points and UT to match more than the first two moments. Moreover, less complicated and more stable versions of UKF with reduced computational cost have also been proposed [51]. C. Ensemble Kalman Filter

The EnKF [52] is a Kalman variant for systems where the state is composed of a large number of variables. This type of problem is encountered frequently in geophysical and hydrological fields, ocean and atmospheric modeling, and data analysis [53]. As discussed in [54] and Section V-E, PFs are not appropriate for problems of high dimensionality. The EnKF is a KF that includes MC sampling. It is possible to argue that the EnKF is a hybrid between a KF and a PF (see Section IV for details on PFs). The EnKF assumes that the large number of state variables and all pdfs are Gaussian, hence the problem is based on a Kalman framework. However, working

(12)

A. Importance Sampling

Before presenting the details of the PF, we summarize how IS and MC integration work. IS is a method employed to compute expectations with respect to one density using random samples drawn from another. Assume that we want to compute an inte. One way of computing is using samplers gral [56] after assuming is a random variable, defining , and rewriting in the form of an expectation

(13)

where is some function of with pdf . This integral can be computed numerically via MC integration [56], by independent and identically distributed samples drawing from the sampling or proposal density

(14)

This method is commonly referred to as the “perfect MC” [40]. This is an unbiased integral estimator, which converges to the true value with diminishing error variance as the number of increases. Another important propparticles drawn from erty of perfect MC is that the convergence is independent of the state dimension, a critical property that IS and many other sampling schemes lack. In many cases, it is too costly or not possible to sample from . Therefore, selection of an appropriate sampling pdf is the critical part in IS. It is customary to use a pdf easy to sample from, such as a Gaussian or uniform. However, in Bayesian tracking and inversion, the integral is already in the form of (13), where is a complex pdf, e.g., (12). It is still possible to use

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a simple function as the sampling density by rewriting (13) and using MC integration to get

TABLE IV SIR PARTICLE FILTER

(15) The estimate is


(16)

with

where

(17)

are the importance weights. The variance is [56]

Expanding the full posterior pdf (only the results shown here; for additional details see [41]), we can write

(18)

The variance in the estimate is minimum if is proportional and increases as deviates from the latter. Using to : as easy to sample IS requires the selection of a balanced from as possible without sacrificing the accuracy of the method.

(24)

where the weight of the th particle at step is computed by inserting (23) and (24) into (22) and ignoring the constant term

(25)

B. Sequential Importance Sampling

Bayesian filtering requires performing successive IS runs at each . The output of each IS run is used as the prior for the next one. This process is referred to as sequential importance samand pling (SIS). Recalling that from Section II-A, it is possible to obtain the from the full posterior density filtering pdf

A key issue in PF design is choosing a good proposal density. , reducing A simple choice is (26) to

(19)

(27)

(20)

Note that, while the full posterior pdf is used in at step deriving IS (19) and (27), only the likelihood is employed in updating weights . The implementation of PF is only concerned with the marginal pdf , as seen in Table IV.

which is in the same form as (13). Assuming a sampling density and performing IS, we obtain (21) (22)

However, to perform IS sequentially at step , we utilize the IS , that is, we use the cloud of results of the previous step particles and associated weights . This is done by selecting (23)

(26)

C. Sequential Importance Resampling

Although SIS provides a complete framework for performing sequential Bayesian estimation, its implementation quickly runs into the problem of sampling degeneracy. After a few iterations of successive SIS, the process leads to a cloud containing few particles with large weights and numerous particles with negligible weights. In the extreme case, there will be only one particle left with a large . This loss of sample diversity results in poor filter performance. To counter degeneracy, a second sampling stage is proposed right after the update stage [57]. The purpose of this resampling

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Fig. 3. The need for resampling. (a) Particles before and after sampling. (b) Particles before resampling and their weights .

refor

stage [58] is to create more high-weight particles from the original set of particles . The modified filter is called SIR [57], [59], [60] and is the most popular PF implementation. Resampling is easily performed at the end of each step . Alternatively, resampling is implemented when the effective needed to maintain diversity drops number of particles below a threshold [61]. The effective number of particles can be estimated as (28)

The resampling process takes the posterior filtering pdf represented by the particle set of SIS at the end of each step and redistributes samples so that all weights of the new particles are the same, that is, . This results in a larger number of particles in the high-likelihood regions, preventing degeneracy. It is best to examine the degeneracy and resampling via an example. 1) Example—Need for Resampling: Resampling is demonin Figs. 3 strated for a SIR PF tracking a target at step 200 particles for state variable and 4. The filter has representing target location. Target location is assumed to be between 0 and 35 with true value . The state equa, where is zero mean tion is: Gaussian noise. (The example is discussed in more detail in Section IV-E1.) that predict . During Fig. 3(a) shows 200 particles the update stage, weights for those particles are computed employing a suitable likelihood function. Only four particles have significant weights and provide substantial information on the posterior pdf; see Fig. 3(b). Resampling is needed for the , so that SIS at generation of more particles around will not be affected by degeneracy. step Fig. 4 illustrates the implementation of the resampling stage. Resampling creates a new discrete density by first computing

Fig. 4. (a) Resampling via the cdf. (b) Histogram of particles after resampling showing large number of newly created particles at high-likelihood regions.

from the cumulative distribution function (cdf) of ; see Fig. 4(a). Using a sample-drawing technique that randomly picks samples from a uniform density, new particles are generated via the inverse cdf. Regions with large weights are replicated multiple times in the new cloud and particles with small weights are less likely to be selected. The are much more focused than new particles after resampling the original set of particles; see Fig. 3(a). The histogram of resampled particles in Fig. 4(b) provides the new estimate of via the likelihood function. Almost 90% of the particles are now in the high-likelihood region. 2) SIR Algorithm: We can merge the SIS and resampling to create the SIR algorithm outlined as follows. Predict: This stage is a SIS prediction that starts with the cloud of equal weigh particles from the previous step , and creates a new set of predictions for the current by sampling from the transitional density step . This is done by propagating each through the state (1) together with a random realization from . This step corresponds to the implementation of (5); see Table IV. for all because of Update: Note that . Having measured , the weight of resampling at step each particle is reevaluated and normalized via (27)

(29)

is the likelihood function defined by the where measurement (2), which includes statistical behavior of errors in the data. The weights of (29) are used in (21) for expressing . the posterior pdf are drawn Resample: New particles from a discrete approximation to density obtained at the update stage. This is achieved through the calculation of



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Fig. 5. Particle flow for a SIR algorithm with

10 particles.

the cdf of as explained above. Different types of resampling are presented in [39] and [62]. All particle weights are now . equal to

A single iteration of a ten-particle SIR algorithm is illustrated in Fig. 5. A cylinder with height proportional to the likelihood represents each particle. As usual, the first stage is prediction, where new particles are created from the particles representing , the state noise at the pdf of the previous step, step , and (1). The likelihood is calculated for each of these new particles using new data . Resampling follows this update stage, where a new set of particles is formed from the previous one. The larger the weight of a particle, the more new particles it generates during resampling. Resampling sometimes adds significantly to the computational requirements of a PF. In this case, quasi-random resampling methods can be used to decrease computational cost as an alternative to the method explained in Fig. 4. The commonly used efficient resampling algorithm is systematic resampling [40]. In this approach, only one random sample uniform distribution and the cdf is is picked from a sliced with equidistant points with spacing starting from the random sample. Original particles are then converted to new particles using the inverse cdf as shown on the right in Fig. 5. Cylinders representing each particle are stacked to obtain the cdf. The new particles are selected by randomly and equally sampling upwards. Further discussion choosing about resampling algorithms is given in Section V-F. , an initial To implement the filtering process for probability density for state vector has to be selected. Prior can be obtained via a sampler. Alternatively, a uniform or Gaussian pdf can be used. The latter requires more steps before stable estimates are obtained. One can sample from to

. Particles in this cloud will generate the initial cloud be propagated through the predict, update, resample process for the estimation of the unknown parameters at the subsequent step. The process at step is summarized in Table IV. D. Advanced Particle Filters

Although SIR provides robust tracking performance in many nonlinear/non-Gaussian tracking problems, it has two basic flaws. 1) Sample impoverishment: Since the resampling stage creates many exact replicas of high-likelihood particles, it is possible to lose sample diversity as the filter progresses. 2) Sensitivity to outliers: The popularity of the SIR filter lies in the simplicity of the selection of transition prior as the sampling density. This is, at the same time, a weakness, because the prediction at step does not use the newly available information in measurement , performing a “blind” prediction. The filter becomes vulnerable to outliers, especially when the likelihood is peaked relative to the prior [41]. To address these problems, advanced PF variants have been developed. [35], [37], [41]. Common ones are briefly discussed in Appendix II in terms of the specific concern they address, their main advantages, and possible drawbacks. Comparative information is provided in Table V, summarizing salient features and differences. E. Sequential Model Selection

Inversion in ocean acoustics typically assumes a known number of parameters that need to be estimated. In geoacoustic inversion, an acoustic propagation model is selected that relates sound to a specific number of environmental parameters. Hence, the model is fixed and known. It was recognized early


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TABLE V FILTER COMPARISON. L: LINEAR G: GAUSSIAN NL: NONLINEAR NG: NON-GAUSSIAN

on that the assumption of a fixed model is not optimal [67]. This observation has lead to efforts towards nonsequential model selection in ocean acoustics with approaches such as hypothesis testing [68], Akaike and Bayesian (Schwartz–Rissanen) information criteria [68]–[70], and evidence calculation [15], [71]. A reversible jump MCMC, where the MC can switch between different models, seems most promising and efficient. For an early implementation, see [72], with an extensive derivation and description in [73]. In dynamic problems, the model may be unknown and changing for each step, similarly to other unknown parameters. Therefore, a sequential state-space system may need more than one model as new data become available. In ocean acoustics, this problem may be encountered in: • tracking an unknown number of targets [74]; • tracking targets with different motion models such as slow moving and fast moving targets, targets with large accelerations, maneuvering targets; • tracking a changing number of unknown vertical modes of the acoustic field, the number of which changes because of range dependence; tracking an unknown number of modes in time-frequency representations; • switching between ray tracing, normal mode, adiabatic normal mode, near-field normal mode, narrow-angle parabolic equation, and wide-angle parabolic equation models that relate data to state parameters, depending on factors such as distance and environmental complexity between the source and the receiver; • geoacoustic tracking of sediments and their varying properties with an unknown number of layers; • arrival time tracking across phones.

Successful tracking under such circumstances requires sequential filtering algorithms that are capable of jumping or switching between models [63], [75]. These filters are referred to as the multiple model particle filters (MMPFs) [41]. The state , becoming , space is augmented with model order takes discrete values corresponding to where parameter and different models. The particles contain samples both for . are updated using values and At each step, the model order probability transition matrix, creating a mixture of particles that use different models. Once model order paramwithin the th particle is determined, a “model-condieter tioned” SIR is run, where the particles, after updates imposed by the state equation, are propagated through the likelihood, relating measurements and state [41]. Since each model has a different number of parameters, the length of the state vector varies with state. Because MMPFs can jump or switch between different models, they are referred to as jump Markov or switching filters. 1) Example: Multiple-Model Target Tracking: The example of Section IV-C is expanded to include a second target that ap. Thus, the single-target model is assigned pears at step and the two-target model uses . the model order An important task is the selection of a transition probability transitions matrix, which determines how model order . Here into

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where it is assumed that the probability of a sudden appearance of a second target or disappearance of an existing target is only

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10% at each time step. Each particle will have its own model . order of order include uncorThe state equation for indices related Gaussian noise with (31) includes adThe measurement equation for observations with and ditive, Gaussian noise (32)


and the size of measurement vector is where 100. The vector has the value of one at index and zero determines how many targets are otherwise. Model order and represent locations of present. We assume that two targets at step . The target locations are functions of step : and . When both targets are present, . True state vector values are shown in Fig. 6(a) (solid lines). and at To estimate state parameters, particles for are drawn from a uniform distribution between 1 and 100. The model order for each particle is initialized via a uniform prior allowing discrete values 1 and 2. Particles are propagated through the state equation, are updated via the measurement equation and resampled, and are then perturbed to predict the following step. Circles in Fig. 6(a) illustrate the modes of the posterior pdfs for state variables. The first target is tracked accurately with small deviations from true values because of noise . The second target is missed at and but is subsequently detected and correctly localized. Fig. 6(b)–(e) demonstrates the posterior pdfs of model order at four consecutive steps. At , only one target is present and the model order pdf clearly supports that. A second but the pdf of the model order remains target enters at , concentrated at 1 and the second target is missed. At the system still favors one target, however, significant proba. At , the presence of two bility appears at . The system learns the presence of targets is favored: the second target slowly because of the low probability of transi. Changing tion between models and the posterior pdf from alters filter behavior. A with a larger transition probability would be more sensitive to sudden changes, capturing the but the location estimates would have larger change at RMS errors since more particles would be allowed to “jump.”

V. PRACTICAL ISSUES

Sequential MC techniques provide a powerful framework for performing signal processing in nonstationary dynamic systems involving nonlinear equations and non-Gaussian pdfs. Implementation of such methods requires a careful selection of a suitable and computationally efficient filter, noise in state and measurement equations, and the number of particles when a particle filtering approach is the best choice. Some of the practical issues arising in state-space estimation are discussed below.

Fig. 6. (a) Target trajectories with a varying model order: true (solid lines) and . Posterior pdf of model order for (b) ; (c) estimated with a PF ; and (e) . ; (d)

A. Expressing the Problem in a Dynamic State-Space Format

Establishing the measurement (2) that relates data to the state vector and the state (1) that expresses the dynamic evois the first step in the implementation of a Bayesian lution of is often the acoustic field received filter. In ocean acoustics, at an array of hydrophones. Some examples of states in ocean acoustics are source bearing, location, velocity and acceleration, ocean bathymetry and sound-speed profile (SSP), geoacoustic parameters, parameters related to acoustic modes and dispersion, frequency, Doppler shift, and multipath arrival time structure along a receiving array.

B. Likelihood Function

An essential part in implementing a sequential PF is to derive the likelihood used in (6) and (29) from the measurement equais derived based on function. The likelihood function for tion and statistical properties of noise . In ocean acoustic applications, data errors are often considered uncorrelated and Gaussian, giving analytically tractable likelihoods. employed for source localAs an example, measurements ization and geoacoustic inversion contain complex-valued hydrophone data across the array and several frequencies. For a single frequency, a simple measurement equation is of the form (33)

is the known or unknown complex-valued source where is the acoustic propagation model. amplitude and


Assuming noise of the form

to be

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, the likelihood function is

(34)

Fig. 7. A quick guide to filter selection.


In geoacoustic tracking, the complex-valued source amplitude is usually unknown. The PF likelihood formulation with an unknown source signal [33] follows closely that of the classical geoacoustic inversion likelihood obtained from the Bartlett power objective function [76]. In most cases, likelihoods derived for inversion can also be used in the sequential filtering. C. State Vector Transitions

divergence performance than the other filter types for particular sets of problems. An example where a PF implementation is required is the problem of atmospheric environmental tracking [77]. The relationship between data and state vector is complex and requires a strongly nonlinear measurement equation with multiple nonadreceived at a naval ditive, non-Gaussian noise terms. Signal consists radar is a function of the changing sea surface and of the atmospheric conditions to be tracked. Data are related to as follows:

State noise in (1) determines the error in the state vector evolution and is often chosen as additive and Gaussian. Furthermore, parameters in the state vector are usually assumed uncoris diagonal. related, i.e., Although state variables are not expected to vary in a way that is not consistent with the state evolution model (1), sudden changes can occur. The filters in such cases need a noise term with a suitable covariance to continue tracking even when the state behaves in an unexpected way. For example, when bathymetry is tracked in a relatively flat ocean environment, the water-column depth varies slowly with range. Hence, small perturbations of the state vector can capture these variations. The presence of a canyon, on the other hand, results in a radical bathymetry change. A small variance in the state noise may then cause the filtering process to diverge. Covariance mashould contain values large enough to accommodate the trix unexpected changes, but at the same time small enough to prevent noisy estimates and poor performance. These tradeoffs can make state noise selection a challenging practical problem. An example where state noise and the number of particles need to be carefully selected because of sharp changes in sediment thickness and sound speed is discussed in Section VI-B.

where is the field calculated using a split-step Fourier and transform parabolic equation for inhomogeneous and represent a compound -distributed pdf (a gamma pdf modulated by a Rayleigh) and a complex Gaussian pdf, respectively. Neither noise component is additive. Even after and reducing to an additive Gaussian eliminating term through a series of approximations, the PF significantly outperforms the EKF and the UKF [29], [78]. This example illustrates a case where the additive Gaussian noise assumption in Section V-B is not suitable.

D. Filter Selection

E. Number of PF Particles

Once an estimation problem is defined with appropriate state and measurement equations, a suitable filter needs to be identified; see Fig. 7. For linear and , additive Gaussian noise terms, the KF is the optimal sequential filter. When and are weakly nonlinear, EKF and UKF provide near-optimal solutions. The EKF has been a popular choice under such circumstances, especially when Jacobian computations of and are straightforward. However, the UKF is superior to the EFK because of higher accuracy in moment estimation and derivative-free implementation using comparable computational resources. EnKFs are recommended for high dimensionality of the state vector. When the estimation problem is too complex to be handled via either analytical or statistical linearization, a PF approach is mandated. However, PFs require considerably more computational resources. PFs are not superior to KFs but rather complementary to them. All variants of Bayesian filters have their place and may perform better in terms of RMS error and

In a classical inversion, the whole search space needs to be explored. However, in sequential Bayesian filtering, the state vector is assumed to follow the state equation. Thus, utilizing the information from the prior step and the known evolution of the states enables a focused search in the state space. Hence, the PF requires a smaller number of particles at each step than the number of samples necessary for a single inversion via, for example, Metropolis–Hastings sampling [14]. The essential number of particles for the filtering process to is problem dependent. successfully estimate posterior . There are four main factors that determine • Problem complexity: Nonlinearities in the state and measurement equations, complexity of the underlying posterior pdf, errors in the state and measurement equations, and de. It takes fewer sired filter output affect the selection of particles to track the median/MAP than the marginal pdf. Highly non-Gaussian, long tailed, peaked, multimodal pdfs differs require a large number of particles. If the state

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depending on whether the Jacobians are calculated analytically sigma points, and or numerically. The UKF needs particles. PFs are critically affected by EnKFs and PFs use the number of particles and the implementation of resampling. Three stages contribute to the computational cost of a SIR particles. The prediction stage uses compuPF with tations to obtain . This is followed by likelihood and weight calculations during the update stage, which are followed by a resampling stage. The resampling algorithm described in the example of Section IV-C is called multinomial [57]. Less computationally expensive resampling algorithms have been proposed such as that operate on the order of residual resampling [84] and systematic resampling [40]. The systematic resampling shown in Fig. 5 is the commonly used resampling algorithm. Overall, the computational cost of a PF is of the order of . Although the complexity of filtering methods is expressed as a function of the number of particles drawn within each step, it is important to consider the computational cost imposed by the propagation model in the measurement equation. In many ocean acoustic applications, the number of operations required for propagation model calculations is significant relative to the filtering. This should be taken into account when evaluating the computational efficiency of sequential filtering approaches. Selecting an algorithm that requires few propagation model calculations without affecting track quality is important.


from the evolution in (1), the actual state will deviate from . A larger is then needed for tracking. predicted • Accuracy: Desired accuracy in track estimates is a major factor in selecting the number of particles. Increasing the number of particles initially provides a large performance improvement in a PF. It has been shown in [79] that the . However, after a problemerror is of the order of dependent is reached, the performance stays relatively flat and increasing the computational cost provides only marginal benefits. An example in geoacoustic tracking is given in [29]. is the state • State dimension: A major factor determining dimension . MC integration is independent of the state statistically independent particles. Howdimension for ever, for sequential filtering, the resampling stage creates multiple copies of high-likelihood particles, making tightly related to the number of state variables [80]. The problem is further complicated by the fact that the importance density in SIR determines convergence. A good samincreasing linearly with [81]. pling density results in increases exponentially with when The required the importance density is poorly chosen (often referred to as “the curse of dimensionality”). Special circumstances facilitate the reduction of the effective size of the state dimension without compromising accuracy. Rao–Blackwellized PFs [82] do just that and are briefly discussed in Appendix II. Note that the effects of a high state dimension are of importance for EnKF applications [54]. on is determined • Computational cost: An upper limit for by the maximum number of particles that can be processed with limited computational resources, which is important especially for real-time filters. schemes [83]. It is possible to have filters with adaptive Coupling PFs with error metrics, the sample size can be adapted depending on the error estimate. Such an implementation is particularly important during the initial steps, when there is only little prior information for . To address the significant uncertainty, a large number of particles can be sampled, with the cloud size decreasing in subsequent steps. F. Computational Cost

Computational cost is related to the overall central processing unit (CPU) time and number of operations a filter has to perform at each step. KFs are generally computationally efficient. EnKFs and PFs can be computationally intensive. The cost is problem specific, depending in particular on the computational complexity of the state transition function , the propagation model , and the ability to sample from the noise pdfs. Following the example in Section V-D, obtaining a Gaussian sample is much simpler than obtaining a -distributed sample. Similarly, the propagation model may require linear matrix multiplications or a complex, nonlinear parabolic equation acoustic propagation model. The KF has the lowest possible computational cost. The computational complexity of KFs and their variations mainly lie in the inversion of matrices necessary for the calculation of the Kalman gain. The EKF needs only slightly more computation,

VI. OCEAN ACOUSTIC APPLICATIONS

Signal processing in ocean acoustics frequently involves complicated and rapidly changing ocean environments in coastal shallow water. The dynamics that characterize ocean acoustic applications are inherently nonlinear, non-Gaussian, and nonstationary processes that quickly vary with space and time. Sequential filtering is able to adapt to these environmental changes. The introduction of newer filters such as UKFs, EnKFs, and PFs that handle nonlinear/non-Gaussian problems, coupled with an increase in computational power, enabled a steady increase in the number of KF and PF approaches developed for ocean acoustic applications. This section is composed of four ocean acoustic examples where tracking filters have been used. A. Target Localization and Tracking With Filtering and Ocean Acoustics

Sequential filters have been applied extensively to target tracking [41]. There is a fair amount of literature about autonomous underwater vehicle (AUV) positioning, navigation, and target tracking [35], [85]. EKFs are frequently employed in target tracking in radar and sonar applications and used in all references cited below. These algorithms involve a state equation that describes the motion of the target in addition to ocean acoustic observations contained in the measurement equation. The state vector may consist of target range, azimuth, coordinates, velocity, and acceleration depending on the complexity of the target motion [35].


shallow-water propagation with reflecting boundaries. The proposed invariance constraint enforces consistency between frequency and range and smoothness in frequency variations with time along an identified striation. Adding the invariance constraint to the EKF improves tracking. B. Environmental Parameter Estimation, Geoacoustic Inversion, and Tracking The relation between the complex ocean environment and acoustic wave propagation involves nonlinear processes. An acoustic propagation model such as normal modes or the parabolic equation is needed depending on the problem. Due to complexity and nonlinearities, environmental sequential Bayesian inversion applications require filters that can handle such problems. Advances in filtering techniques using physics-based environmental inversion are shown in a series of papers discussed below. Candy and Sullivan [19] give an early application of environmental Bayesian filtering, where an EKF-based normal mode code is used [18] to estimate the ocean SSP from noisy acoustic measurements at a single frequency. The EKF marches in depth, constructing the mode shapes together with the SSP as a function of depth. They then apply the algorithm to the Hudson Canyon experiment data [20] and generalize it for broadband processing [23]. These early applications use an EKF. More advanced KFs such as the UKF, the EnKF, and PFs enable estimation and tracking in more complex environments varying temporally and/or spatially both in range and depth. Carriere et al. [28] use the UKF to perform range-independent SSP prediction enabling the temporal tracking of the profile as shown in Fig. 8(a). The SSP is constructed using empirical orthogonal functions (EOFs). The coefficients of each EOF are tracked in time using both an EKF and an UKF. The results confirm that a UKF in range-independent SSP tracking is superior to the EKF. However, the SSP in [28] is a range-independent profile. Temporally tracking a range-dependent SSP requires a much larger number of state parameters to estimate the evolution of the SSP in both depth and range. To solve this problem, Carriere et al. [30] replace the UKF with an EnKF that is efficient for larger state spaces as discussed in Section III-C. They develop an EnKF processor for tracking the range-dependent sound-speed structure from multifrequency sound transmission between a point source and a vertical array. The ocean sound speed is parameterized in terms of range-dependent EOF coefficients. The acoustic measurement equation is augmented by sea-surface sound-speed measurements. They demonstrate the approach from a 13-day experiment in the Ligurian Sea for a 20-km-long transect in a water depth of about 100 m. The introduction of the EnKF decreases the track RMS error significantly and reduces track divergence. A 4-D spatio–temporal environmental tracking of the SSP using full-field tomography is also given in [27] using the EKF. Another application of Bayesian filtering is in geoacoustic inversion and tracking. Just like the water-column SSP, bottom parameters such as SSPs, thickness, attenuation, and density of sedimentary layers affect acoustic propagation. Yardim et al. [29] show how to track these geoacoustic parameters using both vertical and horizontal array data as shown in Fig. 8(b). The


Due to the difficulty of incorporating the complex ocean physics in the estimation process, early work in tracking involved simple ocean models that progressively became more sophisticated. This evolution can be seen in the following series of papers. Culver and Hodgkiss [16] apply an EKF to track the position of an array of freely drifting floats. They use a constant velocity track model for the float positions. The measurements consist of observed travel times between the floats. The ocean acoustic model is responsible for separating the direct arrivals from the surface- and bottom-reflected arrivals. Instead of filtering out the reflected arrivals, El-Hawary and Mbamalu [17] create a more complete ocean model, relating the difference in travel times between direct and reflected arrivals to target range and depth, and solve the tracking problem using various techniques, including EKFs. A unique technique is implemented by Candy and Sullivan [18], [22] in that the propagation model (wave equation) is integrated into both the state and measurement equations, as opposed to using acoustic propagation model just in the measurement equation, common to other ocean acoustic applications. This enables realistic ocean modeling by computing the acoustic field in a normal mode framework. In acoustics, a full propagation model is implemented by marching the field in depth (e.g., a normal mode solution of the Helmholtz equation) or range (e.g., parabolic equation). In [18] and [22], the state vector consists of the modal functions and their horizontal wave numbers and the state equation marches in depth based on a measured SSP. The measurement equation relates the measured complexvalued acoustic pressure on a vertical array to the vector (solutions of the wave equation). This method solves normal mode propagation as an EKF and enables passive source localization in range and depth. Hence, coupling a nonlinear optimization algorithm with the EKF-based ocean acoustic model can solve the source localization problem in a complex ocean environment [21]. Bayesian filtering can track the bearing of a target. Sullivan et al. [25] improve target bearing tracking by applying an EKF to a broadband acoustic field measured on a short array towed by an autonomous submerged vehicle. The moving array causes a Doppler shift in the received signal frequency, which is a function of both bearing and source frequency. By casting the problem as a joint bearing-frequency tracking problem, they track bearing and the rate of change of bearing together with source frequency. An alternative Bayesian tracking approach has been explored extensively [86], [87] for tracking a source in the presence of ocean and geoacoustic variability. Normally, tracking is performed sequentially using SMC. In this paper, the track is parameterized in terms of an analytic equation with unknown parameters (e.g., a straight line with unknown slope) and estimated using nonsequential MC methods. Environmental parameters are included in the inversion to obtain the full posterior pdf. One difference is that they are constant and hence are not tracked. He et al. [26] track with an EKF a target in a horizontal coordinate system illuminated with an active source and a 2-D planar receiver array. The state parameters are the position velocity. The tracking stability is improved by utilizing and the waveguide invariant [88], which is approximately one for

13


example, represent frequencies of an observed signal, related to the measured data in a nonlinear fashion. The unknown number of frequencies also dictates a PF implementation rather than that of a KF or KF variant. Zorych and Michalopoulou [31] track group velocity dispersion curves using PFs, utilizing a normal mode code for sound propagation. Each modal dispersion curve is treated as a “moving target,” whose “location” (here frequency) evolves with time. Synthetic signals at receiving phones from a 20-km range in a 120-m waveguide are processed with an STFT, the magnitude of which is expressed as a weighted sum of Gaussian pulses representing the modes present in the signal. This parametric formulation provides the measurement equation for the PF framework. Frequencies and modal amplitudes are the state parameters. Geoacoustic inversion can employ modal arrival times at specific frequencies as observed data. The PF method described above can enhance inversion by computing uncertainty on extracted arrival times, which can be propagated through the inversion process for the calculation of uncertainty in geoacoustic estimates [92]. Compiling particle information of a specific frequency in each mode provides a posterior pdf of arrival times for this frequency. Instead of estimating wave dispersion using time-frequency representations, Candy and Chambers [24] assume a known dispersion relation in a time-varying waveguide. Under this assumption, their goal is to obtain accurate estimates of the propagating wave from noisy field measurements. This problem is addressed in a state-space framework. The modal wave number is the state variable and the state equation describes the wave number evolution in time between consecutive states. The measurement equation relates field measurements to wave number via the dispersion relation. State transition and measurement functions are nonlinear and an EKF is proposed, calculating Jacobians for the two functions. At each step, MAP wave number estimates are employed in the calculation of an enhanced acoustic field through the measurement equation. Wave number tracking and internal wave estimation of synthetic data demonstrates the approach. Dispersion aside, frequency line tracking has been successfully implemented in other ocean acoustic problems as well. For example, KFs are employed in [93] for the extraction of striations from active sonar spectrograms. This striation identification is useful in verifying target tracks provided by active sonar measurements. Striations relate intensity, target range, and source frequency through the known waveguide invariance [88]. KFs are designed that trace frequency lines in time using smoothness constraints on frequency evolution. The measurement equations relate measured frequency and intensity to the state vector, which also includes frequency and intensity as well as the difference in frequency between consecutive pings.


14

Fig. 8. Geoacoustic environmental tracking. (a) Temporal tracking of the ocean SSP for a fixed receiver and a fixed source. (b) Tracking of the changing environment between the receiver and a moving source. Here shown for a vertical line array (VLA) of receivers.

paper also gives a detailed analysis of performance comparison between the EKF, the UKF, and SIR PFs with changing particle numbers and it was shown the more computationally demanding PF performed best followed by UKF and then EKF. Yardim et al. [33] demonstrate sequential geoacoustic tracking on real data and extend geoacoustic parameter tracking by including the unknown source parameters, effectively tracking the source in an unknown and changing environment. This requires inclusion of the source depth, range, and speed together with the geoacoustic parameters and a source motion model similar to that in target tracking filters. The algorithm is tested on the SWellEX-96[AU: Please provide

full name and location of the experiment]

data [89] with bathymetry ranging from 100 to 250 m and a range-dependent sedimentary layer. C. Frequency Tracking

Time-frequency representations provide a wealth of information on the time-varying evolution of the frequency content of nonstationary signals. Short-time Fourier transforms (STFTs) and time-varying autoregressive (TVAR) methods have been frequently employed in, e.g., audio processing [90] and speech processing [91]. PFs offer an excellent framework for instantaneous frequency extraction in such problems. Data measurements (time slices of the STFT or TVAR spectra) are related to the varying frequencies through a measurement equation. The state vector consists of peak frequencies that vary smoothly between consecutive time steps as given by the state equation. In addition to peak frequencies, the state vector includes the corresponding peak amplitudes and the model order (how many frequency components are present); see Section IV-E for the treatment of a varying or unknown model order. TVAR offers different choices for state-space formulation, with the state vector including either autoregressive coefficients or poles and moduli. Poles, for

D. Spatial Arrival Time Tracking

In ocean acoustics, multipath arrival times have been successfully used for source location, bathymetry estimation, and geoacoustic inversion [17], [94]. This approach is especially helpful for short-range, high-frequency sound propagation in a



shallow-water waveguide. Such circumstances allow us to observe and quantify interactions of sound paths with the propagation medium and boundaries. The quality of source localization and environmental parameter inversion depends on the accuracy of arrival time estimates. A PF was employed for time-delay estimation [95], where arrival times at two receivers were tracked in time for signals propagating in a time-varying environment. The measurement equation related measurements to arrival times and the state equation allowed limited and controlled variations between time states. In addition to evolution in time, sequential Bayesian filtering is employed to trace spatial variations of arrival times. In [32], a PF extracted accurate travel times (state variables) from acoustic signals (measurements). Spatial rather than temporal sequential variability of arrivals was exploited. Following the process of [32], we applied the PF arrival time estimation approach to data from the 2006 Shallow Water Experiment (SW06). The data were received on August 27, 2006, at the 16-element MPL-VLA1 vertical array, just before geoacoustic inversion was performed [96], [97]. A linear-frequency-modulated signal was transmitted with frequency content between 100 and 900 Hz; the sampling rate was 50 kHz. The source depth was 25 m and the range to VLA was 230 m. Received signals were match filtered to produce the time series of Fig. 9(a). Only the 14 lower phones were used here, as the SNR at the top two phones was low due to large ocean waves. The purpose is to estimate the arrival times of three paths at each phone (direct path and surface and bottom reflections), because accurate estimation is needed for inversion based on extracted path arrivals. The state variables for the PF are the arrival times for the three paths of Fig. 9(a). Similarly to the example of Section IV-E, both state and measurement equations and , respectively. include uncorrelated Gaussian noise The state equation is given in (31). The measurement equation is

15

(35)

The pulse results from the autocorrelation of the linearly frequency-modulated source waveform of bandwidth , which was match filtered similarly to the received time series. The likelihood function for the unknown arrival times at receiver is

(36)

Data covariance matrix is assumed diagonal with all . Quantities are amplitudes nonzero elements equal to of the arriving paths that are here assumed to be known. The sequential filtering is started from the top (phone 14). Initially, is assumed to be the same for all phones and equal order to three. Fig. 9(b) demonstrates the pdfs of peak arrival times as calculated by the PF. The spread of the pdfs as well as the number of modes express the uncertainty in arrival time estimation. The

Fig. 9. (a) Observed time series at range 230 m from the VLA showing direct, surface-, and bottom-reflected arrivals; (b) pdf of peak arrival time for each of these arrivals.

third arrival is the weakest in terms of amplitude. As a result, the pdf for this arrival at the top phone is spread and bimodal. The uncertainty is reduced at lower phones, where prior information builds and sequentially propagates from one phone to the next. We next expanded the state vector to include the order, namely the number of arrivals that are present in the time could be 1, 2, or 3. We used only a portion series. That is, of the time series of Fig. 9(a), up to 14 ms, excluding the bottom echo from phones 14–9. At the beginning of the filtering process, the prior on the number of arrivals was uniform allowing values 1, 2, and 3; the transition matrix probability between model orders (numbers of arrivals) is given by

(37)

PDFs of peak arrival times at each phone are shown in Fig. 10(a). Two arrivals are detected at phones 14–8. The third arrival appears in the time series at receiver 8. The PF misses it there but captures it at phones 7–1. Due to the sequential nature of the Bayesian estimation process, where the previous estimate influences the current, the approach is biased towards the prior solution. The transition from two to three arrivals in Fig. 10(b)–(d) illustrates the predict-and-update process. The posterior pdf on the order at phone 9 exhibits a significant probability for the presence of two arrivals, which is the actual case. At phone 8, when the third arrival appears, its presence appears plausible, but lags behind the probability for two arrivals, because of the information that is propagated from the previous phone. At phone 7, the order is comfortably estimated to be three, with little probability corresponding to the order of two. That is, after two updating steps, the estimation process ”learns” that an order jump has occurred. Increased uncertainty is evident in the surface- and bottom-reflected paths near their crossing in the neighborhood of the fifth

16


APPENDIX I CHOICE OF POSTERIOR PDF


It is important to identify which posterior pdfs of the unare calculated in problems such as the known state variables ones encountered in ocean acoustics, depending on the available and limitations on complexity. There data measurements may be significant computational differences between these approaches. ; inversion. This is the marginal posterior pdf cal1) culated in ocean acoustics where a state equation is not utilized to predict the subsequent steps. The estimate of does not use information from previous or future measurements. It is computationally inefficient to perform each inversion independently with . Further, the estimates can be inconsistent from step to step. ; full posterior. The dimension of can 2) become large, as the dimension of is the product of number of data measurements and states, making the estimation of the full posterior pdf practically impossible. An example of such an approach in ocean acoustics is has been computed for a given in [86], where in a nonsequential way. small ; filter: Bayesian filtering is the topic of this 3) paper. Filtering enables all the previous and current meausing “predict” and surements to be used in estimating “update” steps. ; smoother. A smoother is appropriate 4) in applications where all data have already been observed and are readily available. Therefore, both past and “future” measurements can be exploited for the calculation of mar. Smoothing is more complex and ginal pdf computationally expensive than filtering. Although inclusion of future data improves estimation in comparison to a one-way filtering approach, the increase in computational cost sometimes makes smoothing less desirable than filtering. ; predictor. It is ideal for applications 5) where the acoustic measurements collected up to and including the current step are needed to predict some future . It is the most common form where value of state is known as the one-step ahead predictor.

Fig. 10. (a) PDF of arrival times for two or three arrivals. Model order is varying with each hydrophone. (b) PDF of model order (number of arrivals) present at phones 9, 8, and 7.

phone. The arrivals can still be well estimated as modes of the computed pdfs, with the direction of the paths being correctly identified after the crossing.

VII. CONCLUSION

The theory for sequential Bayesian filtering of acoustic signals is summarized focusing on classical KF and PFs. Filtering tracks the evolving state parameters as new data become available. It is based on two equations: state equation (transitions between states) and measurement equation (relation between data and states for one measurement). KFs and their extensions are simple in their implementation and computationally efficient. PFs are computationally onerous but make fewer assumptions than the KF family, solve complex problems, and are still straightforward to implement. Hence, KFs and PFs complement each other enabling us to successfully address a wide spectrum of problems involving dynamical systems. Ocean acoustic measurements in many applications are by nature sequential. Thus, the powerful sequential Bayesian filtering approaches can be applied to a broad range of acoustic problems. In the examples that were presented in this paper, we described how classical target tracking has evolved into the more complex problem of environmental tracking. Model selection is another important example as well as frequency component extraction from acoustic measurements. We showed how PFs can be successfully applied to SW06 data for the extraction of multipath arrival times.

APPENDIX II ADVANCED PARTICLE FILTERS

A. Regularized Particle Filters

The SIR disadvantage associated with the resampling stage is a result of the discrete posterior pdf, designated by a sum of delta functions with corresponding particle weights as given in (21). The resampling stage can only select from these discrete values, creating a large number of identical particles. An alternative is to replace the delta functions in (21) with sharp continuous kernel functions [63]. Since the pdf is now continuous, resampling results in nonidentical particles in the high-likelihood regions, regularizing/smoothing the particle distribution.


17

space. Since the Rao–Blackwellization procedure results in a mixture of Kalman and PF sections and reduces the PF state space via marginalization, the filter is known as mixture PF, marginalized PF, or Rao–Blackwellized PF (RBPF) [41]. A detailed analysis of computational cost and estimate quality of RBPF is given in [82].

B. Markov Chain Monte Carlo Particle Filters

E. Linearized Particle Filter

The Markov chain MC PF (MCMC-PF) [60] is similar to the RPF. It is designed to solve the problem of sample impoverishment by inserting an MCMC sampler in the resampling stage. Such samplers are designed to draw samples from complex pdfs. The histogram of MCMC particles can be shown to converge asymptotically to the desired pdf [56], [98]. Similarly to the PRF, the MCMC kernel and sampling create the necessary sample diversity. Moreover, due to the nature of MCMC, the resulting set of particles still satisfies the asymptotic convergence as opposed to the RPF.

Another interesting variation is the linearized PF (LPF). Instead of using the transitional pdf as the IS density, each particle is treated as a KF and the Gaussian pdfs at the output of each KF are employed as the IS density for the predict stage of the LPF. This creates a bank of KFs, one for each particle and a significant increase in the computational cost for each particle. However, by running a KF on the particles of the previous step, the LPF incorporates both the prior and the current measurement in its predict stage similar to APF does. Since the algorithm uses various forms of KFs, each KF effectively performs a local linearization around its particle. Hence, it is also called the local linearization PF [41]. Successful implementations using both EKFs [59] and UKFs [66] show superior performance compared to SIR. Even though the cost for each particle calculation increases, the total number of particles needed to obtain good track performance decreases.


This type of filter is called the regularized PF (RPF) and resolves the sample impoverishment problem. Kernel approximations are not appropriate for large dimensional state spaces and the introduction of a kernel violates the asymptotic convergence of the particle set to the underlying posterior pdf as the particle number increases.

C. Auxiliary Particle Filter

The auxiliary PF (APF), or auxiliary SIR (ASIR), is introduced by Pitt and Shephard [64] to reduce sample impoverishment and prevent the blind prediction of SIR. This is implemented by delaying the resampling stage that would normally follow the update stage in classical SIR. Instead, the resampling is incorporated into the prediction stage of the next step. The APF introduces an auxiliary variable to state vector that with the prediction stage at combines the resampling at . Therefore, particles at the prediction stage are conditioned on the current prior and the likelihood, resulting in a more efficient filter. APF may be adversely affected in the presence of large process noise [41]. D. Rao–Blackwellized Particle Filters

A main drawback of any PF algorithm is the large amount of particles needed to estimate accurately the posterior. Moreover, this already large number increases quickly as the state-space dimension rises, especially when the IS density used in the PF is different from the true pdf. It is, therefore, desirable to reduce the number of unknown parameters that the PF needs to track. This might be possible in a large number of practical problems (target tracking in particular) [35]. In many cases, not all of the unknown parameters in the state space are nonlinear and nonGaussian. These include target/source velocity and acceleration components. If some subset of the state vector can be modeled as linear/Gaussian conditioned on the rest of the parameters, it might be possible to create tracking filters that are both more efficient and accurate than a standard PF. This is done through Rao–Blackwellization. The Rao–Blackwell theorem shows how to transform an arbitrary estimator into an estimator that is optimal relative to a selected criterion, in the MSE sense [45]. In cases where conditional linear/Gaussian assumption holds, one can Rao–Blackwellize a standard PF. This process involves splitting the state space into two parameter sets: linear/Gaussian and nonlinear/non-Gaussian parameters. We can apply the optimal KF for the marginally linear/Gaussian parameters at each step, concentrating the powerful yet costly PF on the reduced, nonlinear/non-Gaussian portion of the state

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Caglar Yardim (S’98–M’07) received the B.S. and the M.S. degrees in electrical engineering from the Middle East Technical University, Ankara, Turkey, in 1998 and 2001, respectively, and the Ph.D. degree in electrical engineering from the University of California San Diego (UCSD), La Jolla, in 2007. Since 2010, he has been a Scientist at the Marine Physical Laboratory, UCSD. His research interests include signal processing, propagation, optimization, modeling, and inversion of electromagnetic and acoustic signals. Dr. Yardim was the recipient of the Best Student Paper Award at the 2007 IEEE Radar Conference and URSI Young Scientist Award in 2008. He is a member of the Acoustical Society of America.

20


Peter

Gerstoft

received

the

Ph.D.

degree

[AU: Ph.D. in what field of study?]from the Technical University of

Denmark, Lyngby, Denmark, in 1986. From 1987 to 1992, he was with Ødegaard and Danneskiold-Samsøe, Copenhagen, Denmark. From 1992 to 1997, he was at NATO Undersea Research Centre, La Spezia, Italy. Since 1997, he has been with the Marine Physical Laboratory, University of California San Diego, La Jolla. His research interests include modeling and inversion of acoustic, elastic, and electromagnetic signals. Dr. Gerstoft is a Fellow of the Acoustical Society of America and an elected member of the International Union of Radio Science, Commission F.


Zoi-Heleni Michalopoulou (S’85–M’88–SM’02) received the Diploma in electrical engineering from the National Technical University of Athens, Athens, Greece, in 1988 and the M.S. and Ph.D. degrees in electrical engineering from Duke University, Durham, NC, in 1990 and 1993, respectively. Currently, she is a Professor of Mathematical Sciences at the New Jersey Institute of Technology, Newark. Her research interests include underwater acoustics, Bayesian modeling, inverse problems, array signal processing, and spectroscopy. Dr. Michalopoulou is a Fellow of the Acoustical Society of America.


1

An Overview of Sequential Bayesian Filtering in Ocean Acoustics Caglar Yardim, Member, IEEE, Zoi-Heleni Michalopoulou, Senior Member, IEEE, and Peter Gerstoft

Index Terms—Acoustic signal processing, acoustic tracking, ensemble Kalman filter, extended Kalman filter (EKF), ocean acoustics, particle filter (PF), sequential importance resampling (SIR), sequential Monte Carlo methods, unscented Kalman filter (UKF).

I. INTRODUCTION

A

Depending on the data and the problem, Bayesian approaches can be used to address inversion or tracking problems (a brief comparison is given in Appendix I). Sequential Bayesian filtering, tying together information on parameter evolution, a function relating acoustic field measurements to the unknown quantities, and a statistical description of the random perturbations in the field measurements, offers a suitable framework for the solution of such problems. When functions relating 1) parameters between consecutive steps and 2) data and parameters are linear and noise is additive and Gaussian, the Kalman filter (KF) is the optimal estimator in terms of minimizing mean squared error (MSE). The KF propagates expectations and covariances of the unknown parameters from step to step, fully characterizing posterior pdfs. Some early examples used in source tracking in the ocean with KFs are given in [16] and [17]. Nonlinear functions require variations or generalization of the standard filter. Implementation in ocean acoustic problems of a straightforward extension, the extended Kalman filter (EKF), which linearizes mildly nonlinear functions, has been pioneered in a series of papers [18]–[27]. More recently, unscented Kalman filters (UKFs) have achieved better root-mean-square (RMS) error and convergence performance than EKFs by selecting deterministic points called sigma points to represent parameter pdfs [28], [29]. For large parameter vectors, the ensemble KF (EnKF) is efficient [30]. Highly nonlinear systems and complex noise processes require numerical methods for the computation of posterior pdfs. These approaches are termed particle filters (PFs) and have been encountered in ocean acoustic applications in [29] and [31]–[33]. In this tutorial paper, we briefly review the foundations of sequential filters, starting from the well-known KF and its variants and proceeding with PFs. These techniques formulate sequential estimation using physical relations between unknown parameters and measurements embedded in noise environments of a diverse nature. We examine and compare filters and present examples, illustrating practical challenges and solutions. A more detailed presentation of these methods, including theoretical derivations, can be found in many excellent signal processing papers [34]–[39] and texts on sequential Monte Carlo methods [40]–[43]. Once basic principles and methods of sequential filtering are discussed, the focus is shifted to an overview of sequential filtering implemented for parameter estimation of dynamical systems in ocean acoustics. We present applications in target tracking, wave estimation, geoacoustic inversion, frequency, and arrival time tracking. The paper is organized as follows. First, a general background about the state-space formulation for the estimation of evolving

IE E Pr E int P r Ve oo rs f ion

Abstract—Sequential filtering provides a suitable framework for estimating and updating the unknown parameters of a system as data become available. The foundations of sequential Bayesian filtering with emphasis on practical issues are first reviewed covering both Kalman and particle filter approaches. Filtering is demonstrated to be a powerful estimation tool, employing prediction from previous estimates and updates stemming from physical and statistical models that relate acoustic measurements to the unknown parameters. Ocean acoustic applications are then reviewed focusing on source tracking, estimation of environmental parameters evolving in time or space, and frequency tracking. Spatial arrival time tracking is illustrated with 2006 Shallow Water Experiment data.

common feature of inverse problems in ocean acoustics is that underlying physical parameters are estimated from measured acoustic data. Examples include source localization [1]–[4], geoacoustic inversion [5]–[9], and marine mammal signal processing [10]. In a Bayesian framework, prior knowledge and acoustic models are combined with a likelihood function to provide posterior probability density functions (pdfs) of parameters of interest. This formulation was first proposed in source localization [11], [12]. Geoacoustic inversion was subsequently approached in a similar fashion, estimating, in addition to source location, ocean environment parameters and their uncertainties [13]–[15]. Often, such parameters evolve in time or space, with acoustic data arriving online at consecutive steps. Information on parameter value evolution and uncertainty at preceding steps can be invaluable for the determination of future estimates but is frequently ignored.

Manuscript received September 02, 2010; revised November 21, 2010; accepted December 03, 2010. This work was supported by the U.S. Office of Naval Research Code 32, under Grants N00014-09-1-0313, N00015-05-1-0264, N00014-05-1-0262, and N00014-10-1-0073. Associate Editor: R. Chapman. C. Yardim and P. Gerstoft are with the Marine Physical Laboratory, Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093-0238 USA (e-mail: [email protected]; [email protected]). Z.-H. Michalopoulou is with the Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 071028 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JOE.2010.2098810

0364-9059/$26.00 © 2010 IEEE

2


Fig. 1. Sequential Bayesian filtering. From state , state is first pre. As data become available, meadicted via the state (1), providing . surement (2) is employed to update state , providing

In a typical ocean acoustic example, function represents some form of a forward or acoustic propagation model and consists of complex-valued acoustic data measurevector ments across a hydrophone array. A sketch of sequential localization of a moving target as described by (1) and (2) is shown in Fig. 1 with circles representing is the estimate of the target location at step . Term using data obtained up to target location and velocity at . By exploiting knowledge of target poand including step sition and trajectory , a filter first predicts , the target location and velocity at step . The prediction is then upbecome available. dated when new data


parameters in dynamical systems is given in Section II together with the basics of Bayesian filtering. The KF framework is introduced in Section III together with its extensions such as the EKF, the UKF, and the EnKFs. The PFs are presented in Section IV. The filter equations are derived starting from the basic importance sampling (IS) concepts, moving to sequential IS, finally deriving the commonly used PF, often referred to as sequential importance resampling (SIR). Advanced PFs are briefly mentioned in Section IV and are presented more extensively in Appendix II. Practical challenges arising in sequential Bayesian filtering are discussed in Section V and ocean acoustic applications are discussed in Section VI. Although Section VI is dedicated to solving estimation problems in the ocean, specific issues in ocean acoustics as they relate to filtering are discussed throughout the paper. II. BACKGROUND

A. State-Space Model

In dynamical systems, a major goal is to estimate parameters that evolve sequentially with time or space. As data become available, the unknown parameters forming a state vector are estimated sequentially using collective data history and prior knowledge on evolution of the state. Tracking in time or space presents us with problems that can be readily formalized in the following framework. be the measurement vector (for example, pressure Let along a hydrophone array in ocean acoustics) at step and let represent the state vector (for example, location of a source emitting signal ), where . The size of the state vector is . Our goal is to sequentially estimate as data meabecome available. surements Two equations define a state-space model (1) (2)

The state equation (1) describes the evolution or transition of with and assumes that states follow a first-order Markov process. Function is a known function relating the state vector at step to that at step . Variable is the process or state . noise and has a known pdf The measurement equation (or observation equation) (2) reto state vector through a known funclates measurements tion . Variable is the measurement noise with a pdf . and can be additive, multiplicative, or incorpoNoise rated in the state and measurement through complex functions and of and , respectively. In addition to the state vector data , the functions and , and noise components and can also change with . In most ocean acoustic applications, the noise terms are adare independent of . Hence, (1) ditive and functions and and (2) are simplified to (3) (4)

B. Bayesian Filtering

Examining the problem from a Bayesian standpoint, we are rather than simple interested in deriving the posterior pdf for of the state vector is aspoint estimates. The initial pdf sumed to be known; it can be assumed in a common form such as uniform or Gaussian, based on ground-truth information, or estimated using numerical techniques such as a Markov chain Monte Carlo (MCMC) sampler on the initial data [14], [44]. be the set of data observed at the Let be the sequence first steps and let of unknown state vectors. It is desirable to estimate the state . Due to vector and its uncertainty at all steps via pdf the computational cost, the problem is simplified by recursively from , estimating the marginal pdf . rather than estimating joint pdf With available, we can predict through the transition pdf . The transition density . Due to the is determined by the state (1) and noise pdf does first-order Markov chain assumption of , not depend on data . Density can be written as

(5)

When a new measurement becomes available, we can upusing the likelihood of the state vector date the state

(6)


and the prediction pdf of (5). The new estimate of the state vector at is calculated via Bayes theorem

3

TABLE I KALMAN FILTER

(7) The denominator in (7) is the normalizing constant of , termed the evidence, and may be expressed as (8)


contains all information necessary for the Density computation of marginal pdfs of elements of , moments, and modes. Optimal sequential estimators [45], such as the minimum mean squared error (MMSE) estimator, which calculates of the state after data have been the expected value observed, and the maximum a posteriori (MAP) estimator that provides the likely value of the state also after data measurement, can be readily implemented. III. KALMAN FILTERS

The discussion of Section II-A raises the question of how to track optimally an evolving density . Kalman [46] showed that, if a Gaussian distribution can represent this evolving pdf, the optimum Bayesian estimator (one that attains the Bayesian Cramér–Rao lower bound (CRLB) [47] with the MMSE) exists and can be computed analytically. This optimum sequential estimator has been termed the KF. Having a Gaussian posterior at each step implies the following. 1) State and measurement noise must be additive. Noise variand , and the prior must be Gaussian. ables and are known linear functions of the state and mea2) surement vectors. This reduces the system given in (1) and (2) to (9) (10)

where and replace and as matrices representing known linear functions of the state vector . State and measurement noise covariances and are

(11)

Even though the parameters given here are real, it is straightforward to extend these to complex cases. The requirements given above are fairly restrictive, however the resulting filter is straightforward to implement. Because a Gaussian pdf is uniquely and completely defined by its first two moments, mean and covariance, the KF needs only recomputing mean and covariance at step , using , , and the new measurement . This is a two-step procedure, repeated at each . • Predict: This stage predicts the current value of the state given its previous value using (9). This is represented by and .

• Update: This stage updates or corrects the values predicted in the previous stage given the information obtained from measurement . This provides the posterior at represented by and . The resulting set of equations for the linear, Gaussian KF is represents the Kalman gain. given in Table I [46], where A. Extended Kalman Filter

Although elegant and easy to implement, the KF has strict linearity and Gaussian pdf requirements that make it unsuitable for a large class of systems that cannot be characterized by (9) and (10). An obvious way to extend the KF framework is linearizing in (1) and (2) around the current state value. functions and and Gaussian noise When coupled with a Gaussian prior terms, the posterior will remain Gaussian. Thus, only mean and covariance need to be tracked via slightly modified KF equations. The new filter, the EKF [45], [48], linearizes locally the state and measurement equations using the first terms in the Taylor series expansions. For the EKF to perform well, nonlinearities should be small and the underlying densities should be close to Gaussian. However, even under these circumstances, because of the implied approximations, the EKF cannot claim the optimality enjoyed by the KF for linear-Gaussian systems. For large nonlinearities, the mean and covariances will not be mapped correctly as shown in the example of Section III-B. Nevertheless, the EKF has been implemented successfully in numerous applications in areas such as radar and sonar target tracking among others. The EKF algorithm is summarized in Table II [45], [48]. B. Unscented Kalman Filter

The simplicity of the Kalman framework arises from its Gaussian nature. The filter only needs to carry the mean and covariance to the next step. A nonlinear system will disrupt the flow of evolving Gaussian pdfs, because a Gaussian random variable passing through a nonlinear transformation will lose its Gaussian form and solely mean and covariance will not be adequate in fully defining this density. This realization leads to the development of KFs that use sigma points coupled with

4


TABLE III UNSCENTED KALMAN FILTER. ( IS THE th COLUMN OF THE MATRIX SQUARE ROOT [49], [50])


TABLE II EXTENDED KALMAN FILTER

unscented transforms (UTs) [49]. The UT enables the propagation of the mean and covariance through nonlinear functions. KFs that make use of the UT are called UKF [50]. While the EKF enforces linearity through analytic linearization, the UKF enforces a Gaussian distribution while keeping the functional nonlinearity. This means that the filter assumes that any nonlinear transformation of a Gaussian input pdf will always be another Gaussian pdf. Since, under this assumption, a Gaussian input results in a Gaussian output, the system is effectively linearized in a statistical rather than analytical sense. This still enables the filter to carry all necessary information by propagating only the mean and covariance as required by the KF. In the UT, a number of weighted sigma points are carefully chosen so that they capture the mean and covariance of the state variable (the sigma points depend on the matrix square root of covariance ; see Table III). As the random variable undergoes a nonlinear transformation, these points are propagated through this nonlinear function and are used to reconstruct the new mean and covariance using the UT weights [49]. Hence, unlike with the EKF, mean and covariance to at least second order of the nonlinearity (third if the initial pdf is Gaussian) can be computed accurately. sigma The UKF uses a recursive formulation where points and their corresponding weights are generated. These are used with the UT to perform the mean and covariance calculations required in the Kalman framework. The UT weights are computed using three independent parameters: prior knowledge parameter , scaling parameter , and parameter , which controls the spread of the sigma points around the mean. The second scaling parameter can be com. The algoputed from these parameters: rithm is presented in Table III. Details about the derivation can be found in [49]. Although UKFs are derivative-free, an improvement over EKFs, and efficient relative to more advanced techniques to be

discussed later, they have two disadvantages as demonstrated in the example below. 1) Example: EKF Versus UKF Versus PF: Assume that a goes Gaussian random variable with with through a nonlinear transformation

A cloud of 10 000 particles drawn from is plotted in Fig. 2 together with the mean and covariance ellipse. When these particles are propagated through the nonlinear equation, the distribution becomes non-Gaussian as shown. The analytical linearization in EKF uses the Jacobian

This will result in a Gaussian density with a mean and covariance of and represented by and


5

with a large covariance matrix in a high-dimensional system is computationally inefficient for the classical KF. This drawback is addressed by using an ensemble of points, similar to the particle set in a PF, to replace the covariance matrix with the ensemble covariance matrix using MC sampling. For detailed theory and practical implementation, see [55]. IV. PARTICLE FILTERS


Fig. 2. Mean and covariance propagation of the EKF and UKF in a nonlinear, non-Gaussian system with . True mean , means from analytical , and statistical linearizations are given with covariance estimates in ellipses for the true covariance and the estimates from the Jacobian terms of (EKF) and the UT (UKF).

Sometimes, the dynamics of a problem cannot be adequately captured by the Kalman family of filters, because of the nonlinear structure of state and measurement equations and the nature of the noise. When KFs and their extensions fail, the estimavia filtering posterior pdfs may be possible tion of using numerical sequential MC techniques, commonly referred to as PFs. using a cloud of parPFs track the posterior pdf that evolve with step . All ticles other estimates can be calculated from , including and solutions, the variance in the estimates, and of the th state variable the marginal posterior pdf

a dashed ellipsoid in Fig. 2. Both mean and covariance have high errors in their estimates. Mean and covariance are also obtained by using the sigma point generation and a successive UT given in Table III. The values obtained by UT are very close to their true values with almost identical ellipses. Fig. 2 highlights two main drawbacks of KFs in nonlinear, non-Gaussian problems. The first is the effect of nonlinearity on the accurate estimation of mean and covariance. This example is “too nonlinear” for the EKF but not for the UKF for obtaining good mean and covariance estimates. However, there are many cases where the nonlinearity may be severe and may require an even higher order accuracy than the UKF can provide to correctly capture mean and covariance. The other is that the densities may be too non-Gaussian to be represented using the first two moments, even if those can be cannot be sufcalculated correctly. The non-Gaussian pdf ficiently characterized with mean and covariance alone in this example. In such cases, the only way to accurately propagate complex, non-Gaussian pdfs is to store the cloud of particles that represent the pdf as shown in Fig. 2. This forms the basic philosophy of particle filtering. The UKF is the first and most widely used sigma-point KF. It is possible to select different sigma points and UT to match more than the first two moments. Moreover, less complicated and more stable versions of UKF with reduced computational cost have also been proposed [51]. C. Ensemble Kalman Filter

The EnKF [52] is a Kalman variant for systems where the state is composed of a large number of variables. This type of problem is encountered frequently in geophysical and hydrological fields, ocean and atmospheric modeling, and data analysis [53]. As discussed in [54] and Section V-E, PFs are not appropriate for problems of high dimensionality. The EnKF is a KF that includes MC sampling. It is possible to argue that the EnKF is a hybrid between a KF and a PF (see Section IV for details on PFs). The EnKF assumes that the large number of state variables and all pdfs are Gaussian, hence the problem is based on a Kalman framework. However, working

(12)

A. Importance Sampling

Before presenting the details of the PF, we summarize how IS and MC integration work. IS is a method employed to compute expectations with respect to one density using random samples drawn from another. Assume that we want to compute an inte. One way of computing is using samplers gral [56] after assuming is a random variable, defining , and rewriting in the form of an expectation

(13)

where is some function of with pdf . This integral can be computed numerically via MC integration [56], by independent and identically distributed samples drawing from the sampling or proposal density

(14)

This method is commonly referred to as the “perfect MC” [40]. This is an unbiased integral estimator, which converges to the true value with diminishing error variance as the number of increases. Another important propparticles drawn from erty of perfect MC is that the convergence is independent of the state dimension, a critical property that IS and many other sampling schemes lack. In many cases, it is too costly or not possible to sample from . Therefore, selection of an appropriate sampling pdf is the critical part in IS. It is customary to use a pdf easy to sample from, such as a Gaussian or uniform. However, in Bayesian tracking and inversion, the integral is already in the form of (13), where is a complex pdf, e.g., (12). It is still possible to use

6


a simple function as the sampling density by rewriting (13) and using MC integration to get

TABLE IV SIR PARTICLE FILTER

(15) The estimate is


(16)

with

where

(17)

are the importance weights. The variance is [56]

Expanding the full posterior pdf (only the results shown here; for additional details see [41]), we can write

(18)

The variance in the estimate is minimum if is proportional and increases as deviates from the latter. Using to : as easy to sample IS requires the selection of a balanced from as possible without sacrificing the accuracy of the method.

(24)

where the weight of the th particle at step is computed by inserting (23) and (24) into (22) and ignoring the constant term

(25)

B. Sequential Importance Sampling

Bayesian filtering requires performing successive IS runs at each . The output of each IS run is used as the prior for the next one. This process is referred to as sequential importance samand pling (SIS). Recalling that from Section II-A, it is possible to obtain the from the full posterior density filtering pdf

A key issue in PF design is choosing a good proposal density. , reducing A simple choice is (26) to

(19)

(27)

(20)

Note that, while the full posterior pdf is used in at step deriving IS (19) and (27), only the likelihood is employed in updating weights . The implementation of PF is only concerned with the marginal pdf , as seen in Table IV.

which is in the same form as (13). Assuming a sampling density and performing IS, we obtain (21) (22)

However, to perform IS sequentially at step , we utilize the IS , that is, we use the cloud of results of the previous step particles and associated weights . This is done by selecting (23)

(26)

C. Sequential Importance Resampling

Although SIS provides a complete framework for performing sequential Bayesian estimation, its implementation quickly runs into the problem of sampling degeneracy. After a few iterations of successive SIS, the process leads to a cloud containing few particles with large weights and numerous particles with negligible weights. In the extreme case, there will be only one particle left with a large . This loss of sample diversity results in poor filter performance. To counter degeneracy, a second sampling stage is proposed right after the update stage [57]. The purpose of this resampling

7



Fig. 3. The need for resampling. (a) Particles before and after sampling. (b) Particles before resampling and their weights .

refor

stage [58] is to create more high-weight particles from the original set of particles . The modified filter is called SIR [57], [59], [60] and is the most popular PF implementation. Resampling is easily performed at the end of each step . Alternatively, resampling is implemented when the effective needed to maintain diversity drops number of particles below a threshold [61]. The effective number of particles can be estimated as (28)

The resampling process takes the posterior filtering pdf represented by the particle set of SIS at the end of each step and redistributes samples so that all weights of the new particles are the same, that is, . This results in a larger number of particles in the high-likelihood regions, preventing degeneracy. It is best to examine the degeneracy and resampling via an example. 1) Example—Need for Resampling: Resampling is demonin Figs. 3 strated for a SIR PF tracking a target at step 200 particles for state variable and 4. The filter has representing target location. Target location is assumed to be between 0 and 35 with true value . The state equa, where is zero mean tion is: Gaussian noise. (The example is discussed in more detail in Section IV-E1.) that predict . During Fig. 3(a) shows 200 particles the update stage, weights for those particles are computed employing a suitable likelihood function. Only four particles have significant weights and provide substantial information on the posterior pdf; see Fig. 3(b). Resampling is needed for the , so that SIS at generation of more particles around will not be affected by degeneracy. step Fig. 4 illustrates the implementation of the resampling stage. Resampling creates a new discrete density by first computing

Fig. 4. (a) Resampling via the cdf. (b) Histogram of particles after resampling showing large number of newly created particles at high-likelihood regions.

from the cumulative distribution function (cdf) of ; see Fig. 4(a). Using a sample-drawing technique that randomly picks samples from a uniform density, new particles are generated via the inverse cdf. Regions with large weights are replicated multiple times in the new cloud and particles with small weights are less likely to be selected. The are much more focused than new particles after resampling the original set of particles; see Fig. 3(a). The histogram of resampled particles in Fig. 4(b) provides the new estimate of via the likelihood function. Almost 90% of the particles are now in the high-likelihood region. 2) SIR Algorithm: We can merge the SIS and resampling to create the SIR algorithm outlined as follows. Predict: This stage is a SIS prediction that starts with the cloud of equal weigh particles from the previous step , and creates a new set of predictions for the current by sampling from the transitional density step . This is done by propagating each through the state (1) together with a random realization from . This step corresponds to the implementation of (5); see Table IV. for all because of Update: Note that . Having measured , the weight of resampling at step each particle is reevaluated and normalized via (27)

(29)

is the likelihood function defined by the where measurement (2), which includes statistical behavior of errors in the data. The weights of (29) are used in (21) for expressing . the posterior pdf are drawn Resample: New particles from a discrete approximation to density obtained at the update stage. This is achieved through the calculation of



8

Fig. 5. Particle flow for a SIR algorithm with

10 particles.

the cdf of as explained above. Different types of resampling are presented in [39] and [62]. All particle weights are now . equal to

A single iteration of a ten-particle SIR algorithm is illustrated in Fig. 5. A cylinder with height proportional to the likelihood represents each particle. As usual, the first stage is prediction, where new particles are created from the particles representing , the state noise at the pdf of the previous step, step , and (1). The likelihood is calculated for each of these new particles using new data . Resampling follows this update stage, where a new set of particles is formed from the previous one. The larger the weight of a particle, the more new particles it generates during resampling. Resampling sometimes adds significantly to the computational requirements of a PF. In this case, quasi-random resampling methods can be used to decrease computational cost as an alternative to the method explained in Fig. 4. The commonly used efficient resampling algorithm is systematic resampling [40]. In this approach, only one random sample uniform distribution and the cdf is is picked from a sliced with equidistant points with spacing starting from the random sample. Original particles are then converted to new particles using the inverse cdf as shown on the right in Fig. 5. Cylinders representing each particle are stacked to obtain the cdf. The new particles are selected by randomly and equally sampling upwards. Further discussion choosing about resampling algorithms is given in Section V-F. , an initial To implement the filtering process for probability density for state vector has to be selected. Prior can be obtained via a sampler. Alternatively, a uniform or Gaussian pdf can be used. The latter requires more steps before stable estimates are obtained. One can sample from to

. Particles in this cloud will generate the initial cloud be propagated through the predict, update, resample process for the estimation of the unknown parameters at the subsequent step. The process at step is summarized in Table IV. D. Advanced Particle Filters

Although SIR provides robust tracking performance in many nonlinear/non-Gaussian tracking problems, it has two basic flaws. 1) Sample impoverishment: Since the resampling stage creates many exact replicas of high-likelihood particles, it is possible to lose sample diversity as the filter progresses. 2) Sensitivity to outliers: The popularity of the SIR filter lies in the simplicity of the selection of transition prior as the sampling density. This is, at the same time, a weakness, because the prediction at step does not use the newly available information in measurement , performing a “blind” prediction. The filter becomes vulnerable to outliers, especially when the likelihood is peaked relative to the prior [41]. To address these problems, advanced PF variants have been developed. [35], [37], [41]. Common ones are briefly discussed in Appendix II in terms of the specific concern they address, their main advantages, and possible drawbacks. Comparative information is provided in Table V, summarizing salient features and differences. E. Sequential Model Selection

Inversion in ocean acoustics typically assumes a known number of parameters that need to be estimated. In geoacoustic inversion, an acoustic propagation model is selected that relates sound to a specific number of environmental parameters. Hence, the model is fixed and known. It was recognized early


9


TABLE V FILTER COMPARISON. L: LINEAR G: GAUSSIAN NL: NONLINEAR NG: NON-GAUSSIAN

on that the assumption of a fixed model is not optimal [67]. This observation has lead to efforts towards nonsequential model selection in ocean acoustics with approaches such as hypothesis testing [68], Akaike and Bayesian (Schwartz–Rissanen) information criteria [68]–[70], and evidence calculation [15], [71]. A reversible jump MCMC, where the MC can switch between different models, seems most promising and efficient. For an early implementation, see [72], with an extensive derivation and description in [73]. In dynamic problems, the model may be unknown and changing for each step, similarly to other unknown parameters. Therefore, a sequential state-space system may need more than one model as new data become available. In ocean acoustics, this problem may be encountered in: • tracking an unknown number of targets [74]; • tracking targets with different motion models such as slow moving and fast moving targets, targets with large accelerations, maneuvering targets; • tracking a changing number of unknown vertical modes of the acoustic field, the number of which changes because of range dependence; tracking an unknown number of modes in time-frequency representations; • switching between ray tracing, normal mode, adiabatic normal mode, near-field normal mode, narrow-angle parabolic equation, and wide-angle parabolic equation models that relate data to state parameters, depending on factors such as distance and environmental complexity between the source and the receiver; • geoacoustic tracking of sediments and their varying properties with an unknown number of layers; • arrival time tracking across phones.

Successful tracking under such circumstances requires sequential filtering algorithms that are capable of jumping or switching between models [63], [75]. These filters are referred to as the multiple model particle filters (MMPFs) [41]. The state , becoming , space is augmented with model order takes discrete values corresponding to where parameter and different models. The particles contain samples both for . are updated using values and At each step, the model order probability transition matrix, creating a mixture of particles that use different models. Once model order paramwithin the th particle is determined, a “model-condieter tioned” SIR is run, where the particles, after updates imposed by the state equation, are propagated through the likelihood, relating measurements and state [41]. Since each model has a different number of parameters, the length of the state vector varies with state. Because MMPFs can jump or switch between different models, they are referred to as jump Markov or switching filters. 1) Example: Multiple-Model Target Tracking: The example of Section IV-C is expanded to include a second target that ap. Thus, the single-target model is assigned pears at step and the two-target model uses . the model order An important task is the selection of a transition probability transitions matrix, which determines how model order . Here into

(30)

where it is assumed that the probability of a sudden appearance of a second target or disappearance of an existing target is only

10


10% at each time step. Each particle will have its own model . order of order include uncorThe state equation for indices related Gaussian noise with (31) includes adThe measurement equation for observations with and ditive, Gaussian noise (32)


and the size of measurement vector is where 100. The vector has the value of one at index and zero determines how many targets are otherwise. Model order and represent locations of present. We assume that two targets at step . The target locations are functions of step : and . When both targets are present, . True state vector values are shown in Fig. 6(a) (solid lines). and at To estimate state parameters, particles for are drawn from a uniform distribution between 1 and 100. The model order for each particle is initialized via a uniform prior allowing discrete values 1 and 2. Particles are propagated through the state equation, are updated via the measurement equation and resampled, and are then perturbed to predict the following step. Circles in Fig. 6(a) illustrate the modes of the posterior pdfs for state variables. The first target is tracked accurately with small deviations from true values because of noise . The second target is missed at and but is subsequently detected and correctly localized. Fig. 6(b)–(e) demonstrates the posterior pdfs of model order at four consecutive steps. At , only one target is present and the model order pdf clearly supports that. A second but the pdf of the model order remains target enters at , concentrated at 1 and the second target is missed. At the system still favors one target, however, significant proba. At , the presence of two bility appears at . The system learns the presence of targets is favored: the second target slowly because of the low probability of transi. Changing tion between models and the posterior pdf from alters filter behavior. A with a larger transition probability would be more sensitive to sudden changes, capturing the but the location estimates would have larger change at RMS errors since more particles would be allowed to “jump.”

V. PRACTICAL ISSUES

Sequential MC techniques provide a powerful framework for performing signal processing in nonstationary dynamic systems involving nonlinear equations and non-Gaussian pdfs. Implementation of such methods requires a careful selection of a suitable and computationally efficient filter, noise in state and measurement equations, and the number of particles when a particle filtering approach is the best choice. Some of the practical issues arising in state-space estimation are discussed below.

Fig. 6. (a) Target trajectories with a varying model order: true (solid lines) and . Posterior pdf of model order for (b) ; (c) estimated with a PF ; and (e) . ; (d)

A. Expressing the Problem in a Dynamic State-Space Format

Establishing the measurement (2) that relates data to the state vector and the state (1) that expresses the dynamic evois the first step in the implementation of a Bayesian lution of is often the acoustic field received filter. In ocean acoustics, at an array of hydrophones. Some examples of states in ocean acoustics are source bearing, location, velocity and acceleration, ocean bathymetry and sound-speed profile (SSP), geoacoustic parameters, parameters related to acoustic modes and dispersion, frequency, Doppler shift, and multipath arrival time structure along a receiving array.

B. Likelihood Function

An essential part in implementing a sequential PF is to derive the likelihood used in (6) and (29) from the measurement equais derived based on function. The likelihood function for tion and statistical properties of noise . In ocean acoustic applications, data errors are often considered uncorrelated and Gaussian, giving analytically tractable likelihoods. employed for source localAs an example, measurements ization and geoacoustic inversion contain complex-valued hydrophone data across the array and several frequencies. For a single frequency, a simple measurement equation is of the form (33)

is the known or unknown complex-valued source where is the acoustic propagation model. amplitude and


Assuming noise of the form

to be

11

, the likelihood function is

(34)

Fig. 7. A quick guide to filter selection.


In geoacoustic tracking, the complex-valued source amplitude is usually unknown. The PF likelihood formulation with an unknown source signal [33] follows closely that of the classical geoacoustic inversion likelihood obtained from the Bartlett power objective function [76]. In most cases, likelihoods derived for inversion can also be used in the sequential filtering. C. State Vector Transitions

divergence performance than the other filter types for particular sets of problems. An example where a PF implementation is required is the problem of atmospheric environmental tracking [77]. The relationship between data and state vector is complex and requires a strongly nonlinear measurement equation with multiple nonadreceived at a naval ditive, non-Gaussian noise terms. Signal consists radar is a function of the changing sea surface and of the atmospheric conditions to be tracked. Data are related to as follows:

State noise in (1) determines the error in the state vector evolution and is often chosen as additive and Gaussian. Furthermore, parameters in the state vector are usually assumed uncoris diagonal. related, i.e., Although state variables are not expected to vary in a way that is not consistent with the state evolution model (1), sudden changes can occur. The filters in such cases need a noise term with a suitable covariance to continue tracking even when the state behaves in an unexpected way. For example, when bathymetry is tracked in a relatively flat ocean environment, the water-column depth varies slowly with range. Hence, small perturbations of the state vector can capture these variations. The presence of a canyon, on the other hand, results in a radical bathymetry change. A small variance in the state noise may then cause the filtering process to diverge. Covariance mashould contain values large enough to accommodate the trix unexpected changes, but at the same time small enough to prevent noisy estimates and poor performance. These tradeoffs can make state noise selection a challenging practical problem. An example where state noise and the number of particles need to be carefully selected because of sharp changes in sediment thickness and sound speed is discussed in Section VI-B.

where is the field calculated using a split-step Fourier and transform parabolic equation for inhomogeneous and represent a compound -distributed pdf (a gamma pdf modulated by a Rayleigh) and a complex Gaussian pdf, respectively. Neither noise component is additive. Even after and reducing to an additive Gaussian eliminating term through a series of approximations, the PF significantly outperforms the EKF and the UKF [29], [78]. This example illustrates a case where the additive Gaussian noise assumption in Section V-B is not suitable.

D. Filter Selection

E. Number of PF Particles

Once an estimation problem is defined with appropriate state and measurement equations, a suitable filter needs to be identified; see Fig. 7. For linear and , additive Gaussian noise terms, the KF is the optimal sequential filter. When and are weakly nonlinear, EKF and UKF provide near-optimal solutions. The EKF has been a popular choice under such circumstances, especially when Jacobian computations of and are straightforward. However, the UKF is superior to the EFK because of higher accuracy in moment estimation and derivative-free implementation using comparable computational resources. EnKFs are recommended for high dimensionality of the state vector. When the estimation problem is too complex to be handled via either analytical or statistical linearization, a PF approach is mandated. However, PFs require considerably more computational resources. PFs are not superior to KFs but rather complementary to them. All variants of Bayesian filters have their place and may perform better in terms of RMS error and

In a classical inversion, the whole search space needs to be explored. However, in sequential Bayesian filtering, the state vector is assumed to follow the state equation. Thus, utilizing the information from the prior step and the known evolution of the states enables a focused search in the state space. Hence, the PF requires a smaller number of particles at each step than the number of samples necessary for a single inversion via, for example, Metropolis–Hastings sampling [14]. The essential number of particles for the filtering process to is problem dependent. successfully estimate posterior . There are four main factors that determine • Problem complexity: Nonlinearities in the state and measurement equations, complexity of the underlying posterior pdf, errors in the state and measurement equations, and de. It takes fewer sired filter output affect the selection of particles to track the median/MAP than the marginal pdf. Highly non-Gaussian, long tailed, peaked, multimodal pdfs differs require a large number of particles. If the state

12


depending on whether the Jacobians are calculated analytically sigma points, and or numerically. The UKF needs particles. PFs are critically affected by EnKFs and PFs use the number of particles and the implementation of resampling. Three stages contribute to the computational cost of a SIR particles. The prediction stage uses compuPF with tations to obtain . This is followed by likelihood and weight calculations during the update stage, which are followed by a resampling stage. The resampling algorithm described in the example of Section IV-C is called multinomial [57]. Less computationally expensive resampling algorithms have been proposed such as that operate on the order of residual resampling [84] and systematic resampling [40]. The systematic resampling shown in Fig. 5 is the commonly used resampling algorithm. Overall, the computational cost of a PF is of the order of . Although the complexity of filtering methods is expressed as a function of the number of particles drawn within each step, it is important to consider the computational cost imposed by the propagation model in the measurement equation. In many ocean acoustic applications, the number of operations required for propagation model calculations is significant relative to the filtering. This should be taken into account when evaluating the computational efficiency of sequential filtering approaches. Selecting an algorithm that requires few propagation model calculations without affecting track quality is important.


from the evolution in (1), the actual state will deviate from . A larger is then needed for tracking. predicted • Accuracy: Desired accuracy in track estimates is a major factor in selecting the number of particles. Increasing the number of particles initially provides a large performance improvement in a PF. It has been shown in [79] that the . However, after a problemerror is of the order of dependent is reached, the performance stays relatively flat and increasing the computational cost provides only marginal benefits. An example in geoacoustic tracking is given in [29]. is the state • State dimension: A major factor determining dimension . MC integration is independent of the state statistically independent particles. Howdimension for ever, for sequential filtering, the resampling stage creates multiple copies of high-likelihood particles, making tightly related to the number of state variables [80]. The problem is further complicated by the fact that the importance density in SIR determines convergence. A good samincreasing linearly with [81]. pling density results in increases exponentially with when The required the importance density is poorly chosen (often referred to as “the curse of dimensionality”). Special circumstances facilitate the reduction of the effective size of the state dimension without compromising accuracy. Rao–Blackwellized PFs [82] do just that and are briefly discussed in Appendix II. Note that the effects of a high state dimension are of importance for EnKF applications [54]. on is determined • Computational cost: An upper limit for by the maximum number of particles that can be processed with limited computational resources, which is important especially for real-time filters. schemes [83]. It is possible to have filters with adaptive Coupling PFs with error metrics, the sample size can be adapted depending on the error estimate. Such an implementation is particularly important during the initial steps, when there is only little prior information for . To address the significant uncertainty, a large number of particles can be sampled, with the cloud size decreasing in subsequent steps. F. Computational Cost

Computational cost is related to the overall central processing unit (CPU) time and number of operations a filter has to perform at each step. KFs are generally computationally efficient. EnKFs and PFs can be computationally intensive. The cost is problem specific, depending in particular on the computational complexity of the state transition function , the propagation model , and the ability to sample from the noise pdfs. Following the example in Section V-D, obtaining a Gaussian sample is much simpler than obtaining a -distributed sample. Similarly, the propagation model may require linear matrix multiplications or a complex, nonlinear parabolic equation acoustic propagation model. The KF has the lowest possible computational cost. The computational complexity of KFs and their variations mainly lie in the inversion of matrices necessary for the calculation of the Kalman gain. The EKF needs only slightly more computation,

VI. OCEAN ACOUSTIC APPLICATIONS

Signal processing in ocean acoustics frequently involves complicated and rapidly changing ocean environments in coastal shallow water. The dynamics that characterize ocean acoustic applications are inherently nonlinear, non-Gaussian, and nonstationary processes that quickly vary with space and time. Sequential filtering is able to adapt to these environmental changes. The introduction of newer filters such as UKFs, EnKFs, and PFs that handle nonlinear/non-Gaussian problems, coupled with an increase in computational power, enabled a steady increase in the number of KF and PF approaches developed for ocean acoustic applications. This section is composed of four ocean acoustic examples where tracking filters have been used. A. Target Localization and Tracking With Filtering and Ocean Acoustics

Sequential filters have been applied extensively to target tracking [41]. There is a fair amount of literature about autonomous underwater vehicle (AUV) positioning, navigation, and target tracking [35], [85]. EKFs are frequently employed in target tracking in radar and sonar applications and used in all references cited below. These algorithms involve a state equation that describes the motion of the target in addition to ocean acoustic observations contained in the measurement equation. The state vector may consist of target range, azimuth, coordinates, velocity, and acceleration depending on the complexity of the target motion [35].


shallow-water propagation with reflecting boundaries. The proposed invariance constraint enforces consistency between frequency and range and smoothness in frequency variations with time along an identified striation. Adding the invariance constraint to the EKF improves tracking. B. Environmental Parameter Estimation, Geoacoustic Inversion, and Tracking The relation between the complex ocean environment and acoustic wave propagation involves nonlinear processes. An acoustic propagation model such as normal modes or the parabolic equation is needed depending on the problem. Due to complexity and nonlinearities, environmental sequential Bayesian inversion applications require filters that can handle such problems. Advances in filtering techniques using physics-based environmental inversion are shown in a series of papers discussed below. Candy and Sullivan [19] give an early application of environmental Bayesian filtering, where an EKF-based normal mode code is used [18] to estimate the ocean SSP from noisy acoustic measurements at a single frequency. The EKF marches in depth, constructing the mode shapes together with the SSP as a function of depth. They then apply the algorithm to the Hudson Canyon experiment data [20] and generalize it for broadband processing [23]. These early applications use an EKF. More advanced KFs such as the UKF, the EnKF, and PFs enable estimation and tracking in more complex environments varying temporally and/or spatially both in range and depth. Carriere et al. [28] use the UKF to perform range-independent SSP prediction enabling the temporal tracking of the profile as shown in Fig. 8(a). The SSP is constructed using empirical orthogonal functions (EOFs). The coefficients of each EOF are tracked in time using both an EKF and an UKF. The results confirm that a UKF in range-independent SSP tracking is superior to the EKF. However, the SSP in [28] is a range-independent profile. Temporally tracking a range-dependent SSP requires a much larger number of state parameters to estimate the evolution of the SSP in both depth and range. To solve this problem, Carriere et al. [30] replace the UKF with an EnKF that is efficient for larger state spaces as discussed in Section III-C. They develop an EnKF processor for tracking the range-dependent sound-speed structure from multifrequency sound transmission between a point source and a vertical array. The ocean sound speed is parameterized in terms of range-dependent EOF coefficients. The acoustic measurement equation is augmented by sea-surface sound-speed measurements. They demonstrate the approach from a 13-day experiment in the Ligurian Sea for a 20-km-long transect in a water depth of about 100 m. The introduction of the EnKF decreases the track RMS error significantly and reduces track divergence. A 4-D spatio–temporal environmental tracking of the SSP using full-field tomography is also given in [27] using the EKF. Another application of Bayesian filtering is in geoacoustic inversion and tracking. Just like the water-column SSP, bottom parameters such as SSPs, thickness, attenuation, and density of sedimentary layers affect acoustic propagation. Yardim et al. [29] show how to track these geoacoustic parameters using both vertical and horizontal array data as shown in Fig. 8(b). The


Due to the difficulty of incorporating the complex ocean physics in the estimation process, early work in tracking involved simple ocean models that progressively became more sophisticated. This evolution can be seen in the following series of papers. Culver and Hodgkiss [16] apply an EKF to track the position of an array of freely drifting floats. They use a constant velocity track model for the float positions. The measurements consist of observed travel times between the floats. The ocean acoustic model is responsible for separating the direct arrivals from the surface- and bottom-reflected arrivals. Instead of filtering out the reflected arrivals, El-Hawary and Mbamalu [17] create a more complete ocean model, relating the difference in travel times between direct and reflected arrivals to target range and depth, and solve the tracking problem using various techniques, including EKFs. A unique technique is implemented by Candy and Sullivan [18], [22] in that the propagation model (wave equation) is integrated into both the state and measurement equations, as opposed to using acoustic propagation model just in the measurement equation, common to other ocean acoustic applications. This enables realistic ocean modeling by computing the acoustic field in a normal mode framework. In acoustics, a full propagation model is implemented by marching the field in depth (e.g., a normal mode solution of the Helmholtz equation) or range (e.g., parabolic equation). In [18] and [22], the state vector consists of the modal functions and their horizontal wave numbers and the state equation marches in depth based on a measured SSP. The measurement equation relates the measured complexvalued acoustic pressure on a vertical array to the vector (solutions of the wave equation). This method solves normal mode propagation as an EKF and enables passive source localization in range and depth. Hence, coupling a nonlinear optimization algorithm with the EKF-based ocean acoustic model can solve the source localization problem in a complex ocean environment [21]. Bayesian filtering can track the bearing of a target. Sullivan et al. [25] improve target bearing tracking by applying an EKF to a broadband acoustic field measured on a short array towed by an autonomous submerged vehicle. The moving array causes a Doppler shift in the received signal frequency, which is a function of both bearing and source frequency. By casting the problem as a joint bearing-frequency tracking problem, they track bearing and the rate of change of bearing together with source frequency. An alternative Bayesian tracking approach has been explored extensively [86], [87] for tracking a source in the presence of ocean and geoacoustic variability. Normally, tracking is performed sequentially using SMC. In this paper, the track is parameterized in terms of an analytic equation with unknown parameters (e.g., a straight line with unknown slope) and estimated using nonsequential MC methods. Environmental parameters are included in the inversion to obtain the full posterior pdf. One difference is that they are constant and hence are not tracked. He et al. [26] track with an EKF a target in a horizontal coordinate system illuminated with an active source and a 2-D planar receiver array. The state parameters are the position velocity. The tracking stability is improved by utilizing and the waveguide invariant [88], which is approximately one for

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example, represent frequencies of an observed signal, related to the measured data in a nonlinear fashion. The unknown number of frequencies also dictates a PF implementation rather than that of a KF or KF variant. Zorych and Michalopoulou [31] track group velocity dispersion curves using PFs, utilizing a normal mode code for sound propagation. Each modal dispersion curve is treated as a “moving target,” whose “location” (here frequency) evolves with time. Synthetic signals at receiving phones from a 20-km range in a 120-m waveguide are processed with an STFT, the magnitude of which is expressed as a weighted sum of Gaussian pulses representing the modes present in the signal. This parametric formulation provides the measurement equation for the PF framework. Frequencies and modal amplitudes are the state parameters. Geoacoustic inversion can employ modal arrival times at specific frequencies as observed data. The PF method described above can enhance inversion by computing uncertainty on extracted arrival times, which can be propagated through the inversion process for the calculation of uncertainty in geoacoustic estimates [92]. Compiling particle information of a specific frequency in each mode provides a posterior pdf of arrival times for this frequency. Instead of estimating wave dispersion using time-frequency representations, Candy and Chambers [24] assume a known dispersion relation in a time-varying waveguide. Under this assumption, their goal is to obtain accurate estimates of the propagating wave from noisy field measurements. This problem is addressed in a state-space framework. The modal wave number is the state variable and the state equation describes the wave number evolution in time between consecutive states. The measurement equation relates field measurements to wave number via the dispersion relation. State transition and measurement functions are nonlinear and an EKF is proposed, calculating Jacobians for the two functions. At each step, MAP wave number estimates are employed in the calculation of an enhanced acoustic field through the measurement equation. Wave number tracking and internal wave estimation of synthetic data demonstrates the approach. Dispersion aside, frequency line tracking has been successfully implemented in other ocean acoustic problems as well. For example, KFs are employed in [93] for the extraction of striations from active sonar spectrograms. This striation identification is useful in verifying target tracks provided by active sonar measurements. Striations relate intensity, target range, and source frequency through the known waveguide invariance [88]. KFs are designed that trace frequency lines in time using smoothness constraints on frequency evolution. The measurement equations relate measured frequency and intensity to the state vector, which also includes frequency and intensity as well as the difference in frequency between consecutive pings.


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Fig. 8. Geoacoustic environmental tracking. (a) Temporal tracking of the ocean SSP for a fixed receiver and a fixed source. (b) Tracking of the changing environment between the receiver and a moving source. Here shown for a vertical line array (VLA) of receivers.

paper also gives a detailed analysis of performance comparison between the EKF, the UKF, and SIR PFs with changing particle numbers and it was shown the more computationally demanding PF performed best followed by UKF and then EKF. Yardim et al. [33] demonstrate sequential geoacoustic tracking on real data and extend geoacoustic parameter tracking by including the unknown source parameters, effectively tracking the source in an unknown and changing environment. This requires inclusion of the source depth, range, and speed together with the geoacoustic parameters and a source motion model similar to that in target tracking filters. The algorithm is tested on the SWellEX-96[AU: Please provide

full name and location of the experiment]

data [89] with bathymetry ranging from 100 to 250 m and a range-dependent sedimentary layer. C. Frequency Tracking

Time-frequency representations provide a wealth of information on the time-varying evolution of the frequency content of nonstationary signals. Short-time Fourier transforms (STFTs) and time-varying autoregressive (TVAR) methods have been frequently employed in, e.g., audio processing [90] and speech processing [91]. PFs offer an excellent framework for instantaneous frequency extraction in such problems. Data measurements (time slices of the STFT or TVAR spectra) are related to the varying frequencies through a measurement equation. The state vector consists of peak frequencies that vary smoothly between consecutive time steps as given by the state equation. In addition to peak frequencies, the state vector includes the corresponding peak amplitudes and the model order (how many frequency components are present); see Section IV-E for the treatment of a varying or unknown model order. TVAR offers different choices for state-space formulation, with the state vector including either autoregressive coefficients or poles and moduli. Poles, for

D. Spatial Arrival Time Tracking

In ocean acoustics, multipath arrival times have been successfully used for source location, bathymetry estimation, and geoacoustic inversion [17], [94]. This approach is especially helpful for short-range, high-frequency sound propagation in a



shallow-water waveguide. Such circumstances allow us to observe and quantify interactions of sound paths with the propagation medium and boundaries. The quality of source localization and environmental parameter inversion depends on the accuracy of arrival time estimates. A PF was employed for time-delay estimation [95], where arrival times at two receivers were tracked in time for signals propagating in a time-varying environment. The measurement equation related measurements to arrival times and the state equation allowed limited and controlled variations between time states. In addition to evolution in time, sequential Bayesian filtering is employed to trace spatial variations of arrival times. In [32], a PF extracted accurate travel times (state variables) from acoustic signals (measurements). Spatial rather than temporal sequential variability of arrivals was exploited. Following the process of [32], we applied the PF arrival time estimation approach to data from the 2006 Shallow Water Experiment (SW06). The data were received on August 27, 2006, at the 16-element MPL-VLA1 vertical array, just before geoacoustic inversion was performed [96], [97]. A linear-frequency-modulated signal was transmitted with frequency content between 100 and 900 Hz; the sampling rate was 50 kHz. The source depth was 25 m and the range to VLA was 230 m. Received signals were match filtered to produce the time series of Fig. 9(a). Only the 14 lower phones were used here, as the SNR at the top two phones was low due to large ocean waves. The purpose is to estimate the arrival times of three paths at each phone (direct path and surface and bottom reflections), because accurate estimation is needed for inversion based on extracted path arrivals. The state variables for the PF are the arrival times for the three paths of Fig. 9(a). Similarly to the example of Section IV-E, both state and measurement equations and , respectively. include uncorrelated Gaussian noise The state equation is given in (31). The measurement equation is

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(35)

The pulse results from the autocorrelation of the linearly frequency-modulated source waveform of bandwidth , which was match filtered similarly to the received time series. The likelihood function for the unknown arrival times at receiver is

(36)

Data covariance matrix is assumed diagonal with all . Quantities are amplitudes nonzero elements equal to of the arriving paths that are here assumed to be known. The sequential filtering is started from the top (phone 14). Initially, is assumed to be the same for all phones and equal order to three. Fig. 9(b) demonstrates the pdfs of peak arrival times as calculated by the PF. The spread of the pdfs as well as the number of modes express the uncertainty in arrival time estimation. The

Fig. 9. (a) Observed time series at range 230 m from the VLA showing direct, surface-, and bottom-reflected arrivals; (b) pdf of peak arrival time for each of these arrivals.

third arrival is the weakest in terms of amplitude. As a result, the pdf for this arrival at the top phone is spread and bimodal. The uncertainty is reduced at lower phones, where prior information builds and sequentially propagates from one phone to the next. We next expanded the state vector to include the order, namely the number of arrivals that are present in the time could be 1, 2, or 3. We used only a portion series. That is, of the time series of Fig. 9(a), up to 14 ms, excluding the bottom echo from phones 14–9. At the beginning of the filtering process, the prior on the number of arrivals was uniform allowing values 1, 2, and 3; the transition matrix probability between model orders (numbers of arrivals) is given by

(37)

PDFs of peak arrival times at each phone are shown in Fig. 10(a). Two arrivals are detected at phones 14–8. The third arrival appears in the time series at receiver 8. The PF misses it there but captures it at phones 7–1. Due to the sequential nature of the Bayesian estimation process, where the previous estimate influences the current, the approach is biased towards the prior solution. The transition from two to three arrivals in Fig. 10(b)–(d) illustrates the predict-and-update process. The posterior pdf on the order at phone 9 exhibits a significant probability for the presence of two arrivals, which is the actual case. At phone 8, when the third arrival appears, its presence appears plausible, but lags behind the probability for two arrivals, because of the information that is propagated from the previous phone. At phone 7, the order is comfortably estimated to be three, with little probability corresponding to the order of two. That is, after two updating steps, the estimation process ”learns” that an order jump has occurred. Increased uncertainty is evident in the surface- and bottom-reflected paths near their crossing in the neighborhood of the fifth

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APPENDIX I CHOICE OF POSTERIOR PDF


It is important to identify which posterior pdfs of the unare calculated in problems such as the known state variables ones encountered in ocean acoustics, depending on the available and limitations on complexity. There data measurements may be significant computational differences between these approaches. ; inversion. This is the marginal posterior pdf cal1) culated in ocean acoustics where a state equation is not utilized to predict the subsequent steps. The estimate of does not use information from previous or future measurements. It is computationally inefficient to perform each inversion independently with . Further, the estimates can be inconsistent from step to step. ; full posterior. The dimension of can 2) become large, as the dimension of is the product of number of data measurements and states, making the estimation of the full posterior pdf practically impossible. An example of such an approach in ocean acoustics is has been computed for a given in [86], where in a nonsequential way. small ; filter: Bayesian filtering is the topic of this 3) paper. Filtering enables all the previous and current meausing “predict” and surements to be used in estimating “update” steps. ; smoother. A smoother is appropriate 4) in applications where all data have already been observed and are readily available. Therefore, both past and “future” measurements can be exploited for the calculation of mar. Smoothing is more complex and ginal pdf computationally expensive than filtering. Although inclusion of future data improves estimation in comparison to a one-way filtering approach, the increase in computational cost sometimes makes smoothing less desirable than filtering. ; predictor. It is ideal for applications 5) where the acoustic measurements collected up to and including the current step are needed to predict some future . It is the most common form where value of state is known as the one-step ahead predictor.

Fig. 10. (a) PDF of arrival times for two or three arrivals. Model order is varying with each hydrophone. (b) PDF of model order (number of arrivals) present at phones 9, 8, and 7.

phone. The arrivals can still be well estimated as modes of the computed pdfs, with the direction of the paths being correctly identified after the crossing.

VII. CONCLUSION

The theory for sequential Bayesian filtering of acoustic signals is summarized focusing on classical KF and PFs. Filtering tracks the evolving state parameters as new data become available. It is based on two equations: state equation (transitions between states) and measurement equation (relation between data and states for one measurement). KFs and their extensions are simple in their implementation and computationally efficient. PFs are computationally onerous but make fewer assumptions than the KF family, solve complex problems, and are still straightforward to implement. Hence, KFs and PFs complement each other enabling us to successfully address a wide spectrum of problems involving dynamical systems. Ocean acoustic measurements in many applications are by nature sequential. Thus, the powerful sequential Bayesian filtering approaches can be applied to a broad range of acoustic problems. In the examples that were presented in this paper, we described how classical target tracking has evolved into the more complex problem of environmental tracking. Model selection is another important example as well as frequency component extraction from acoustic measurements. We showed how PFs can be successfully applied to SW06 data for the extraction of multipath arrival times.

APPENDIX II ADVANCED PARTICLE FILTERS

A. Regularized Particle Filters

The SIR disadvantage associated with the resampling stage is a result of the discrete posterior pdf, designated by a sum of delta functions with corresponding particle weights as given in (21). The resampling stage can only select from these discrete values, creating a large number of identical particles. An alternative is to replace the delta functions in (21) with sharp continuous kernel functions [63]. Since the pdf is now continuous, resampling results in nonidentical particles in the high-likelihood regions, regularizing/smoothing the particle distribution.


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space. Since the Rao–Blackwellization procedure results in a mixture of Kalman and PF sections and reduces the PF state space via marginalization, the filter is known as mixture PF, marginalized PF, or Rao–Blackwellized PF (RBPF) [41]. A detailed analysis of computational cost and estimate quality of RBPF is given in [82].

B. Markov Chain Monte Carlo Particle Filters

E. Linearized Particle Filter

The Markov chain MC PF (MCMC-PF) [60] is similar to the RPF. It is designed to solve the problem of sample impoverishment by inserting an MCMC sampler in the resampling stage. Such samplers are designed to draw samples from complex pdfs. The histogram of MCMC particles can be shown to converge asymptotically to the desired pdf [56], [98]. Similarly to the PRF, the MCMC kernel and sampling create the necessary sample diversity. Moreover, due to the nature of MCMC, the resulting set of particles still satisfies the asymptotic convergence as opposed to the RPF.

Another interesting variation is the linearized PF (LPF). Instead of using the transitional pdf as the IS density, each particle is treated as a KF and the Gaussian pdfs at the output of each KF are employed as the IS density for the predict stage of the LPF. This creates a bank of KFs, one for each particle and a significant increase in the computational cost for each particle. However, by running a KF on the particles of the previous step, the LPF incorporates both the prior and the current measurement in its predict stage similar to APF does. Since the algorithm uses various forms of KFs, each KF effectively performs a local linearization around its particle. Hence, it is also called the local linearization PF [41]. Successful implementations using both EKFs [59] and UKFs [66] show superior performance compared to SIR. Even though the cost for each particle calculation increases, the total number of particles needed to obtain good track performance decreases.


This type of filter is called the regularized PF (RPF) and resolves the sample impoverishment problem. Kernel approximations are not appropriate for large dimensional state spaces and the introduction of a kernel violates the asymptotic convergence of the particle set to the underlying posterior pdf as the particle number increases.

C. Auxiliary Particle Filter

The auxiliary PF (APF), or auxiliary SIR (ASIR), is introduced by Pitt and Shephard [64] to reduce sample impoverishment and prevent the blind prediction of SIR. This is implemented by delaying the resampling stage that would normally follow the update stage in classical SIR. Instead, the resampling is incorporated into the prediction stage of the next step. The APF introduces an auxiliary variable to state vector that with the prediction stage at combines the resampling at . Therefore, particles at the prediction stage are conditioned on the current prior and the likelihood, resulting in a more efficient filter. APF may be adversely affected in the presence of large process noise [41]. D. Rao–Blackwellized Particle Filters

A main drawback of any PF algorithm is the large amount of particles needed to estimate accurately the posterior. Moreover, this already large number increases quickly as the state-space dimension rises, especially when the IS density used in the PF is different from the true pdf. It is, therefore, desirable to reduce the number of unknown parameters that the PF needs to track. This might be possible in a large number of practical problems (target tracking in particular) [35]. In many cases, not all of the unknown parameters in the state space are nonlinear and nonGaussian. These include target/source velocity and acceleration components. If some subset of the state vector can be modeled as linear/Gaussian conditioned on the rest of the parameters, it might be possible to create tracking filters that are both more efficient and accurate than a standard PF. This is done through Rao–Blackwellization. The Rao–Blackwell theorem shows how to transform an arbitrary estimator into an estimator that is optimal relative to a selected criterion, in the MSE sense [45]. In cases where conditional linear/Gaussian assumption holds, one can Rao–Blackwellize a standard PF. This process involves splitting the state space into two parameter sets: linear/Gaussian and nonlinear/non-Gaussian parameters. We can apply the optimal KF for the marginally linear/Gaussian parameters at each step, concentrating the powerful yet costly PF on the reduced, nonlinear/non-Gaussian portion of the state

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Caglar Yardim (S’98–M’07) received the B.S. and the M.S. degrees in electrical engineering from the Middle East Technical University, Ankara, Turkey, in 1998 and 2001, respectively, and the Ph.D. degree in electrical engineering from the University of California San Diego (UCSD), La Jolla, in 2007. Since 2010, he has been a Scientist at the Marine Physical Laboratory, UCSD. His research interests include signal processing, propagation, optimization, modeling, and inversion of electromagnetic and acoustic signals. Dr. Yardim was the recipient of the Best Student Paper Award at the 2007 IEEE Radar Conference and URSI Young Scientist Award in 2008. He is a member of the Acoustical Society of America.

20


Peter

Gerstoft

received

the

Ph.D.

degree

[AU: Ph.D. in what field of study?]from the Technical University of

Denmark, Lyngby, Denmark, in 1986. From 1987 to 1992, he was with Ødegaard and Danneskiold-Samsøe, Copenhagen, Denmark. From 1992 to 1997, he was at NATO Undersea Research Centre, La Spezia, Italy. Since 1997, he has been with the Marine Physical Laboratory, University of California San Diego, La Jolla. His research interests include modeling and inversion of acoustic, elastic, and electromagnetic signals. Dr. Gerstoft is a Fellow of the Acoustical Society of America and an elected member of the International Union of Radio Science, Commission F.


Zoi-Heleni Michalopoulou (S’85–M’88–SM’02) received the Diploma in electrical engineering from the National Technical University of Athens, Athens, Greece, in 1988 and the M.S. and Ph.D. degrees in electrical engineering from Duke University, Durham, NC, in 1990 and 1993, respectively. Currently, she is a Professor of Mathematical Sciences at the New Jersey Institute of Technology, Newark. Her research interests include underwater acoustics, Bayesian modeling, inverse problems, array signal processing, and spectroscopy. Dr. Michalopoulou is a Fellow of the Acoustical Society of America.

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